Page 1 J E E - P h y s i c s E 1 NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CURRENT ELECTRICITY In previous chapters we deal largely with electrostatics that is, with charges at rest. With this chapter we begin to focus on electric currents, that is, charges in motion. ELECTRIC CURRENT Electric charges in motion constitute an electric current. Any medium having practically free electric charges, free to migrate is a conductor of electricity. The electric charge flows from higher potential energy state to lower potential energy state. e – I Positive charge flows from higher to lower potential and negative charge flows from lower to higher. Metals such as gold, silver, copper, aluminium etc. are good conductors. When charge flows in a conductor from one place to the other, then the rate of flow of charge is called electric current (I). When there is a transfer of charge from one point to other point in a conductor, we say that there is an electric current through the area. If the moving charges are positive, the current is in the direction of motion of charge. If they are negative the current is opposite to the direction of motion. If a charge ?Q crosses an area in time ?t then the average electric current through the area, during this time as • Average current I av = Q t ? ? • Instantaneous current t 0 Q dQ I Lim t dt ? ? ? ? ? ? GOLDEN KEY POINTS • Current is a fundamental quantity with dimension [M 0 L 0 T 0 A¹ ] • Current is a scalar quantity with its S ? unit ampere. Ampere : The current through a conductor is said to be one ampere if one coulomb of charge is flowing per second through a cross–section of wire. • The conventional direction of current is the direction of flow of positive charge or applied field. It is opposite to direction of flow of negatively charged electrons. • The conductor remains uncharged when current flows through it because the charge entering at one end per second is equal to charge leaving the other end per second. • For a given conductor current does not change with change in its cross–section because current is simply rate of flow of charge. • If n particles each having a charge q pass per second per unit area then current associated with cross– sectional area A is q nqA t ? ? ? ? ? . • If there are n particles per unit volume each having a charge q and moving with velocity v then current through cross–sectional area A is q nqvA t ? ? ? ? ? • If a charge q is moving in a circle of radius r with speed v then its time period is T = 2 ?r/v. The equivalent current q qv T 2 r ? ? ? ? . JEEMAIN.GURU Page 2 J E E - P h y s i c s E 1 NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CURRENT ELECTRICITY In previous chapters we deal largely with electrostatics that is, with charges at rest. With this chapter we begin to focus on electric currents, that is, charges in motion. ELECTRIC CURRENT Electric charges in motion constitute an electric current. Any medium having practically free electric charges, free to migrate is a conductor of electricity. The electric charge flows from higher potential energy state to lower potential energy state. e – I Positive charge flows from higher to lower potential and negative charge flows from lower to higher. Metals such as gold, silver, copper, aluminium etc. are good conductors. When charge flows in a conductor from one place to the other, then the rate of flow of charge is called electric current (I). When there is a transfer of charge from one point to other point in a conductor, we say that there is an electric current through the area. If the moving charges are positive, the current is in the direction of motion of charge. If they are negative the current is opposite to the direction of motion. If a charge ?Q crosses an area in time ?t then the average electric current through the area, during this time as • Average current I av = Q t ? ? • Instantaneous current t 0 Q dQ I Lim t dt ? ? ? ? ? ? GOLDEN KEY POINTS • Current is a fundamental quantity with dimension [M 0 L 0 T 0 A¹ ] • Current is a scalar quantity with its S ? unit ampere. Ampere : The current through a conductor is said to be one ampere if one coulomb of charge is flowing per second through a cross–section of wire. • The conventional direction of current is the direction of flow of positive charge or applied field. It is opposite to direction of flow of negatively charged electrons. • The conductor remains uncharged when current flows through it because the charge entering at one end per second is equal to charge leaving the other end per second. • For a given conductor current does not change with change in its cross–section because current is simply rate of flow of charge. • If n particles each having a charge q pass per second per unit area then current associated with cross– sectional area A is q nqA t ? ? ? ? ? . • If there are n particles per unit volume each having a charge q and moving with velocity v then current through cross–sectional area A is q nqvA t ? ? ? ? ? • If a charge q is moving in a circle of radius r with speed v then its time period is T = 2 ?r/v. The equivalent current q qv T 2 r ? ? ? ? . JEEMAIN.GURU J E E - P h y s i c s 2 E NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CL ASSIFICATION OF MATERIALS ACCORDING TO CONDUCTIVITY ( i ) C o n d u c t o r In some materials, the outer electrons of each atoms or molecules are only weakly bound to it. These electrons are almost free to move throughout the body of the material and are called free electrons. They are also known as conduction electrons. When such a material is placed in an electric field, the free electrons move in a direction opposite to the field. Such materials are called conductors. ( i i ) I n s u l a t o r Another class of materials is called insulators in which all the electrons are tightly bound to their respective atoms or molecules. Effectively, there are no free electrons. When such a material is placed in an electric field, the electrons may slightly shift opposite to the field but they can’t leave their parent atoms or molecules and hence can’t move through long distances. Such materials are also called dielectrics. ( i i i ) S e m i c o n d u c t o r In semiconductors, the behaviour is like an insulator at low levels of temperature. But at higher temperatures, a small number of electrons are able to free themselves and they respond to the applied electric field. As the number of free electrons in a semiconductor is much smaller than that in a conductor, its behaviour is in between a conductor and an insulator and hence, the name semiconductor. A freed electron in a semiconductor leaves a vacancy in its normal bound position. These vacancies also help in conduction. Behavior of conductor in absence of applied potential difference : In absence of applied potential difference electrons have random motion. The average displacement and average velocity is zero. There is no flow of current due to thermal motion of free electrons in a conductor. The free electrons present in a conductor gain energy from temperature of surrounding and move randomly in the conductor. The speed gained by virtue of temperature is called as thermal speed of an electron 2 rms 1 mv 2 = 3 2 kT So thermal speed v rms = 3kT m where m is mass of electron At room temperature T = 300 K, v rms = 10 5 m/s • Mean free path ? ? : ( ?~10Å) , total distance travelled number of collisions ? ? • Relaxation time : The time taken by an electron between two successive collisions is called as relaxation time ? ? : ( ?~10 –14 s), Relaxation time : ? ? total time taken number of collisions Behavior of conductor in presence of applied potential difference : When two ends of a conductors are joined to a battery then one end is at higher potential and another at lower potential. This produces an electric field inside the conductor from point of higher to lower potential E = V L where V = emf of the battery, L = length of the conductor. . The field exerts an electric force on free electrons causing acceleration of each electron. Acceleration of electron F eE a= m m ? ? ? ? ? JEEMAIN.GURU Page 3 J E E - P h y s i c s E 1 NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CURRENT ELECTRICITY In previous chapters we deal largely with electrostatics that is, with charges at rest. With this chapter we begin to focus on electric currents, that is, charges in motion. ELECTRIC CURRENT Electric charges in motion constitute an electric current. Any medium having practically free electric charges, free to migrate is a conductor of electricity. The electric charge flows from higher potential energy state to lower potential energy state. e – I Positive charge flows from higher to lower potential and negative charge flows from lower to higher. Metals such as gold, silver, copper, aluminium etc. are good conductors. When charge flows in a conductor from one place to the other, then the rate of flow of charge is called electric current (I). When there is a transfer of charge from one point to other point in a conductor, we say that there is an electric current through the area. If the moving charges are positive, the current is in the direction of motion of charge. If they are negative the current is opposite to the direction of motion. If a charge ?Q crosses an area in time ?t then the average electric current through the area, during this time as • Average current I av = Q t ? ? • Instantaneous current t 0 Q dQ I Lim t dt ? ? ? ? ? ? GOLDEN KEY POINTS • Current is a fundamental quantity with dimension [M 0 L 0 T 0 A¹ ] • Current is a scalar quantity with its S ? unit ampere. Ampere : The current through a conductor is said to be one ampere if one coulomb of charge is flowing per second through a cross–section of wire. • The conventional direction of current is the direction of flow of positive charge or applied field. It is opposite to direction of flow of negatively charged electrons. • The conductor remains uncharged when current flows through it because the charge entering at one end per second is equal to charge leaving the other end per second. • For a given conductor current does not change with change in its cross–section because current is simply rate of flow of charge. • If n particles each having a charge q pass per second per unit area then current associated with cross– sectional area A is q nqA t ? ? ? ? ? . • If there are n particles per unit volume each having a charge q and moving with velocity v then current through cross–sectional area A is q nqvA t ? ? ? ? ? • If a charge q is moving in a circle of radius r with speed v then its time period is T = 2 ?r/v. The equivalent current q qv T 2 r ? ? ? ? . JEEMAIN.GURU J E E - P h y s i c s 2 E NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CL ASSIFICATION OF MATERIALS ACCORDING TO CONDUCTIVITY ( i ) C o n d u c t o r In some materials, the outer electrons of each atoms or molecules are only weakly bound to it. These electrons are almost free to move throughout the body of the material and are called free electrons. They are also known as conduction electrons. When such a material is placed in an electric field, the free electrons move in a direction opposite to the field. Such materials are called conductors. ( i i ) I n s u l a t o r Another class of materials is called insulators in which all the electrons are tightly bound to their respective atoms or molecules. Effectively, there are no free electrons. When such a material is placed in an electric field, the electrons may slightly shift opposite to the field but they can’t leave their parent atoms or molecules and hence can’t move through long distances. Such materials are also called dielectrics. ( i i i ) S e m i c o n d u c t o r In semiconductors, the behaviour is like an insulator at low levels of temperature. But at higher temperatures, a small number of electrons are able to free themselves and they respond to the applied electric field. As the number of free electrons in a semiconductor is much smaller than that in a conductor, its behaviour is in between a conductor and an insulator and hence, the name semiconductor. A freed electron in a semiconductor leaves a vacancy in its normal bound position. These vacancies also help in conduction. Behavior of conductor in absence of applied potential difference : In absence of applied potential difference electrons have random motion. The average displacement and average velocity is zero. There is no flow of current due to thermal motion of free electrons in a conductor. The free electrons present in a conductor gain energy from temperature of surrounding and move randomly in the conductor. The speed gained by virtue of temperature is called as thermal speed of an electron 2 rms 1 mv 2 = 3 2 kT So thermal speed v rms = 3kT m where m is mass of electron At room temperature T = 300 K, v rms = 10 5 m/s • Mean free path ? ? : ( ?~10Å) , total distance travelled number of collisions ? ? • Relaxation time : The time taken by an electron between two successive collisions is called as relaxation time ? ? : ( ?~10 –14 s), Relaxation time : ? ? total time taken number of collisions Behavior of conductor in presence of applied potential difference : When two ends of a conductors are joined to a battery then one end is at higher potential and another at lower potential. This produces an electric field inside the conductor from point of higher to lower potential E = V L where V = emf of the battery, L = length of the conductor. . The field exerts an electric force on free electrons causing acceleration of each electron. Acceleration of electron F eE a= m m ? ? ? ? ? JEEMAIN.GURU J E E - P h y s i c s E 3 NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 DRIFT VELOCITY Drift velocity is defined as the velocity with which the free electrons get drifted towards the positive terminal under the effect of the applied external electric field. In addition to its thermal velocity, due to acceleration given by applied electric field, the electron acquires a velocity component in a direction opposite to the direction of the electric field. The gain in velocity due to the applied field is very small and is lost in the next collision. e – I Under the action of electric field : Random motion of an electron with superimposed drift At any given time, an electron has a velocity 1 1 1 v u a ? ? ? ? ? ? , where 1 u ? = the thermal velocity and 1 a ? ? = the velocity acquired by the electron under the influence of the applied electric field. ? 1 = the time that has elapsed since the last collision. Similarly, the velocities of the other electrons are 2 2 2 3 3 3 N N N u a , v u a ,... u a v v ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . The average velocity of all the free electrons in the conductor is equal to the drift velocity d v ? of the free electrons 1 2 3 N d v v v ...v N v ? ? ? ? ? ? ? ? ? 1 1 2 2 N N (u a ) (u a ) ... (u ) N a ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 N 1 2 N ( ... u ) ... a N N u u ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 N ... u 0 N u u ? ? ? ? ? ? ? ? 1 2 N d ... N v a ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? d a v ? ? ? ? – m eE ? ? ? Note : Order of drift velocity is 10 –4 m/s. Relation between current and drift velocity : Let n= number density of free electrons and A= area of cross–section of conductor. Number of free electrons in conductor of length L = nAL, Total charge on these free electrons q neAL ? ? Time taken by drifting electrons to cross conductor d L t v ? ? q current I= t ? ? ? = neAL ? ? ? ? ? ? d v L = neAv d Example Find free electrons per unit volume in a metallic wire of density 10 4 kg/m 3 , atomic mass number 100 and number of free electron per atom is one. S o l uti o n Number of free charge particle per unit volume (n) = volume total particle e rg a ch free total ? No. of free electron per atom means total free electrons = total number of atoms= A W N M M ? So A 23 4 W A 3 W N M M N 6.023 10 10 n d V M 100 10 ? ? ? ? ? ? ? ? ? = 6.023 × 10 28 JEEMAIN.GURU Page 4 J E E - P h y s i c s E 1 NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CURRENT ELECTRICITY In previous chapters we deal largely with electrostatics that is, with charges at rest. With this chapter we begin to focus on electric currents, that is, charges in motion. ELECTRIC CURRENT Electric charges in motion constitute an electric current. Any medium having practically free electric charges, free to migrate is a conductor of electricity. The electric charge flows from higher potential energy state to lower potential energy state. e – I Positive charge flows from higher to lower potential and negative charge flows from lower to higher. Metals such as gold, silver, copper, aluminium etc. are good conductors. When charge flows in a conductor from one place to the other, then the rate of flow of charge is called electric current (I). When there is a transfer of charge from one point to other point in a conductor, we say that there is an electric current through the area. If the moving charges are positive, the current is in the direction of motion of charge. If they are negative the current is opposite to the direction of motion. If a charge ?Q crosses an area in time ?t then the average electric current through the area, during this time as • Average current I av = Q t ? ? • Instantaneous current t 0 Q dQ I Lim t dt ? ? ? ? ? ? GOLDEN KEY POINTS • Current is a fundamental quantity with dimension [M 0 L 0 T 0 A¹ ] • Current is a scalar quantity with its S ? unit ampere. Ampere : The current through a conductor is said to be one ampere if one coulomb of charge is flowing per second through a cross–section of wire. • The conventional direction of current is the direction of flow of positive charge or applied field. It is opposite to direction of flow of negatively charged electrons. • The conductor remains uncharged when current flows through it because the charge entering at one end per second is equal to charge leaving the other end per second. • For a given conductor current does not change with change in its cross–section because current is simply rate of flow of charge. • If n particles each having a charge q pass per second per unit area then current associated with cross– sectional area A is q nqA t ? ? ? ? ? . • If there are n particles per unit volume each having a charge q and moving with velocity v then current through cross–sectional area A is q nqvA t ? ? ? ? ? • If a charge q is moving in a circle of radius r with speed v then its time period is T = 2 ?r/v. The equivalent current q qv T 2 r ? ? ? ? . JEEMAIN.GURU J E E - P h y s i c s 2 E NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CL ASSIFICATION OF MATERIALS ACCORDING TO CONDUCTIVITY ( i ) C o n d u c t o r In some materials, the outer electrons of each atoms or molecules are only weakly bound to it. These electrons are almost free to move throughout the body of the material and are called free electrons. They are also known as conduction electrons. When such a material is placed in an electric field, the free electrons move in a direction opposite to the field. Such materials are called conductors. ( i i ) I n s u l a t o r Another class of materials is called insulators in which all the electrons are tightly bound to their respective atoms or molecules. Effectively, there are no free electrons. When such a material is placed in an electric field, the electrons may slightly shift opposite to the field but they can’t leave their parent atoms or molecules and hence can’t move through long distances. Such materials are also called dielectrics. ( i i i ) S e m i c o n d u c t o r In semiconductors, the behaviour is like an insulator at low levels of temperature. But at higher temperatures, a small number of electrons are able to free themselves and they respond to the applied electric field. As the number of free electrons in a semiconductor is much smaller than that in a conductor, its behaviour is in between a conductor and an insulator and hence, the name semiconductor. A freed electron in a semiconductor leaves a vacancy in its normal bound position. These vacancies also help in conduction. Behavior of conductor in absence of applied potential difference : In absence of applied potential difference electrons have random motion. The average displacement and average velocity is zero. There is no flow of current due to thermal motion of free electrons in a conductor. The free electrons present in a conductor gain energy from temperature of surrounding and move randomly in the conductor. The speed gained by virtue of temperature is called as thermal speed of an electron 2 rms 1 mv 2 = 3 2 kT So thermal speed v rms = 3kT m where m is mass of electron At room temperature T = 300 K, v rms = 10 5 m/s • Mean free path ? ? : ( ?~10Å) , total distance travelled number of collisions ? ? • Relaxation time : The time taken by an electron between two successive collisions is called as relaxation time ? ? : ( ?~10 –14 s), Relaxation time : ? ? total time taken number of collisions Behavior of conductor in presence of applied potential difference : When two ends of a conductors are joined to a battery then one end is at higher potential and another at lower potential. This produces an electric field inside the conductor from point of higher to lower potential E = V L where V = emf of the battery, L = length of the conductor. . The field exerts an electric force on free electrons causing acceleration of each electron. Acceleration of electron F eE a= m m ? ? ? ? ? JEEMAIN.GURU J E E - P h y s i c s E 3 NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 DRIFT VELOCITY Drift velocity is defined as the velocity with which the free electrons get drifted towards the positive terminal under the effect of the applied external electric field. In addition to its thermal velocity, due to acceleration given by applied electric field, the electron acquires a velocity component in a direction opposite to the direction of the electric field. The gain in velocity due to the applied field is very small and is lost in the next collision. e – I Under the action of electric field : Random motion of an electron with superimposed drift At any given time, an electron has a velocity 1 1 1 v u a ? ? ? ? ? ? , where 1 u ? = the thermal velocity and 1 a ? ? = the velocity acquired by the electron under the influence of the applied electric field. ? 1 = the time that has elapsed since the last collision. Similarly, the velocities of the other electrons are 2 2 2 3 3 3 N N N u a , v u a ,... u a v v ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . The average velocity of all the free electrons in the conductor is equal to the drift velocity d v ? of the free electrons 1 2 3 N d v v v ...v N v ? ? ? ? ? ? ? ? ? 1 1 2 2 N N (u a ) (u a ) ... (u ) N a ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 N 1 2 N ( ... u ) ... a N N u u ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 N ... u 0 N u u ? ? ? ? ? ? ? ? 1 2 N d ... N v a ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? d a v ? ? ? ? – m eE ? ? ? Note : Order of drift velocity is 10 –4 m/s. Relation between current and drift velocity : Let n= number density of free electrons and A= area of cross–section of conductor. Number of free electrons in conductor of length L = nAL, Total charge on these free electrons q neAL ? ? Time taken by drifting electrons to cross conductor d L t v ? ? q current I= t ? ? ? = neAL ? ? ? ? ? ? d v L = neAv d Example Find free electrons per unit volume in a metallic wire of density 10 4 kg/m 3 , atomic mass number 100 and number of free electron per atom is one. S o l uti o n Number of free charge particle per unit volume (n) = volume total particle e rg a ch free total ? No. of free electron per atom means total free electrons = total number of atoms= A W N M M ? So A 23 4 W A 3 W N M M N 6.023 10 10 n d V M 100 10 ? ? ? ? ? ? ? ? ? = 6.023 × 10 28 JEEMAIN.GURU J E E - P h y s i c s 4 E NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CURRENT DENSITY (J) Current is a macroscopic quantity and deals with the overall rate of flow of charge through a section. To specify the current with direction in the microscopic level at a point, the term current density is introduced. Current density at any point inside a conductor is defined as a vector having magnitude equal to current per unit area surrounding that point. Remember area is normal to the direction of charge flow (or current passes) through that point. • Current density at point P is given by dI J n dA ? ? ? + I dA n J I ? ? dA J dA cos ? – • If the cross–sectional area is not normal to the current, but makes an angle ? with the direction of current then dI J dA cos ? ? ? dI = JdA cos ? = J.dA ? ? ? I J . dA ? ? ? ? ? ? ? • Current density J ? is a vector quantity. It's direction is same as that of E ? . It's S.I. unit is ampere/m 2 and dimension [L –2 A]. Example The current density at a point is ? ? ? ? ? ? 4 2 ˆ J 2 10 j Jm . Find the rate of charge flow through a cross sectional area ? ? 2 ˆ ˆ S 2i 3 j cm ? ? ? Solution The rate of flow of charge = current = I = J.dS ? ? ? ? I = J.S ? ? = ? ? 4 2 10 ? ? ? 4 ˆ ˆ ˆ j 2i 3 j 10 A 6A ? ? ? ? ? ? ? ? ? Example A potential difference applied to the ends of a wire made up of an alloy drives a current through it. The current density varies as J = 3 + 2r, where r is the distance of the point from the axis. If R be the radius of the wire, then the total current through any cross section of the wire. Solution Consider a circular strip of radius r and thickness dr dI = J.dS ? ? = ? ? ? ? 3 2r 2 rdr cos0 ? ? ? = ? ? 2 2 3r 2r dr ? ? ? ? R 2 0 I 2 3r 2r dr ? ? ? ? = R 2 3 0 3r 2 2 r 2 3 ? ? ? ? ? ? ? ? = 2 3 3R 2R 2 2 3 ? ? ? ? ? ? ? ? units RELATION BETWEEN CURRENT DENSITY, CONDUCTIVITY AND ELECTRIC FIELD Let the number of free electrons per unit volume in a conductor = n Total number of electrons in dx distance = n (Adx) JEEMAIN.GURU Page 5 J E E - P h y s i c s E 1 NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CURRENT ELECTRICITY In previous chapters we deal largely with electrostatics that is, with charges at rest. With this chapter we begin to focus on electric currents, that is, charges in motion. ELECTRIC CURRENT Electric charges in motion constitute an electric current. Any medium having practically free electric charges, free to migrate is a conductor of electricity. The electric charge flows from higher potential energy state to lower potential energy state. e – I Positive charge flows from higher to lower potential and negative charge flows from lower to higher. Metals such as gold, silver, copper, aluminium etc. are good conductors. When charge flows in a conductor from one place to the other, then the rate of flow of charge is called electric current (I). When there is a transfer of charge from one point to other point in a conductor, we say that there is an electric current through the area. If the moving charges are positive, the current is in the direction of motion of charge. If they are negative the current is opposite to the direction of motion. If a charge ?Q crosses an area in time ?t then the average electric current through the area, during this time as • Average current I av = Q t ? ? • Instantaneous current t 0 Q dQ I Lim t dt ? ? ? ? ? ? GOLDEN KEY POINTS • Current is a fundamental quantity with dimension [M 0 L 0 T 0 A¹ ] • Current is a scalar quantity with its S ? unit ampere. Ampere : The current through a conductor is said to be one ampere if one coulomb of charge is flowing per second through a cross–section of wire. • The conventional direction of current is the direction of flow of positive charge or applied field. It is opposite to direction of flow of negatively charged electrons. • The conductor remains uncharged when current flows through it because the charge entering at one end per second is equal to charge leaving the other end per second. • For a given conductor current does not change with change in its cross–section because current is simply rate of flow of charge. • If n particles each having a charge q pass per second per unit area then current associated with cross– sectional area A is q nqA t ? ? ? ? ? . • If there are n particles per unit volume each having a charge q and moving with velocity v then current through cross–sectional area A is q nqvA t ? ? ? ? ? • If a charge q is moving in a circle of radius r with speed v then its time period is T = 2 ?r/v. The equivalent current q qv T 2 r ? ? ? ? . JEEMAIN.GURU J E E - P h y s i c s 2 E NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CL ASSIFICATION OF MATERIALS ACCORDING TO CONDUCTIVITY ( i ) C o n d u c t o r In some materials, the outer electrons of each atoms or molecules are only weakly bound to it. These electrons are almost free to move throughout the body of the material and are called free electrons. They are also known as conduction electrons. When such a material is placed in an electric field, the free electrons move in a direction opposite to the field. Such materials are called conductors. ( i i ) I n s u l a t o r Another class of materials is called insulators in which all the electrons are tightly bound to their respective atoms or molecules. Effectively, there are no free electrons. When such a material is placed in an electric field, the electrons may slightly shift opposite to the field but they can’t leave their parent atoms or molecules and hence can’t move through long distances. Such materials are also called dielectrics. ( i i i ) S e m i c o n d u c t o r In semiconductors, the behaviour is like an insulator at low levels of temperature. But at higher temperatures, a small number of electrons are able to free themselves and they respond to the applied electric field. As the number of free electrons in a semiconductor is much smaller than that in a conductor, its behaviour is in between a conductor and an insulator and hence, the name semiconductor. A freed electron in a semiconductor leaves a vacancy in its normal bound position. These vacancies also help in conduction. Behavior of conductor in absence of applied potential difference : In absence of applied potential difference electrons have random motion. The average displacement and average velocity is zero. There is no flow of current due to thermal motion of free electrons in a conductor. The free electrons present in a conductor gain energy from temperature of surrounding and move randomly in the conductor. The speed gained by virtue of temperature is called as thermal speed of an electron 2 rms 1 mv 2 = 3 2 kT So thermal speed v rms = 3kT m where m is mass of electron At room temperature T = 300 K, v rms = 10 5 m/s • Mean free path ? ? : ( ?~10Å) , total distance travelled number of collisions ? ? • Relaxation time : The time taken by an electron between two successive collisions is called as relaxation time ? ? : ( ?~10 –14 s), Relaxation time : ? ? total time taken number of collisions Behavior of conductor in presence of applied potential difference : When two ends of a conductors are joined to a battery then one end is at higher potential and another at lower potential. This produces an electric field inside the conductor from point of higher to lower potential E = V L where V = emf of the battery, L = length of the conductor. . The field exerts an electric force on free electrons causing acceleration of each electron. Acceleration of electron F eE a= m m ? ? ? ? ? JEEMAIN.GURU J E E - P h y s i c s E 3 NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 DRIFT VELOCITY Drift velocity is defined as the velocity with which the free electrons get drifted towards the positive terminal under the effect of the applied external electric field. In addition to its thermal velocity, due to acceleration given by applied electric field, the electron acquires a velocity component in a direction opposite to the direction of the electric field. The gain in velocity due to the applied field is very small and is lost in the next collision. e – I Under the action of electric field : Random motion of an electron with superimposed drift At any given time, an electron has a velocity 1 1 1 v u a ? ? ? ? ? ? , where 1 u ? = the thermal velocity and 1 a ? ? = the velocity acquired by the electron under the influence of the applied electric field. ? 1 = the time that has elapsed since the last collision. Similarly, the velocities of the other electrons are 2 2 2 3 3 3 N N N u a , v u a ,... u a v v ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . The average velocity of all the free electrons in the conductor is equal to the drift velocity d v ? of the free electrons 1 2 3 N d v v v ...v N v ? ? ? ? ? ? ? ? ? 1 1 2 2 N N (u a ) (u a ) ... (u ) N a ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 N 1 2 N ( ... u ) ... a N N u u ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 N ... u 0 N u u ? ? ? ? ? ? ? ? 1 2 N d ... N v a ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? d a v ? ? ? ? – m eE ? ? ? Note : Order of drift velocity is 10 –4 m/s. Relation between current and drift velocity : Let n= number density of free electrons and A= area of cross–section of conductor. Number of free electrons in conductor of length L = nAL, Total charge on these free electrons q neAL ? ? Time taken by drifting electrons to cross conductor d L t v ? ? q current I= t ? ? ? = neAL ? ? ? ? ? ? d v L = neAv d Example Find free electrons per unit volume in a metallic wire of density 10 4 kg/m 3 , atomic mass number 100 and number of free electron per atom is one. S o l uti o n Number of free charge particle per unit volume (n) = volume total particle e rg a ch free total ? No. of free electron per atom means total free electrons = total number of atoms= A W N M M ? So A 23 4 W A 3 W N M M N 6.023 10 10 n d V M 100 10 ? ? ? ? ? ? ? ? ? = 6.023 × 10 28 JEEMAIN.GURU J E E - P h y s i c s 4 E NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 CURRENT DENSITY (J) Current is a macroscopic quantity and deals with the overall rate of flow of charge through a section. To specify the current with direction in the microscopic level at a point, the term current density is introduced. Current density at any point inside a conductor is defined as a vector having magnitude equal to current per unit area surrounding that point. Remember area is normal to the direction of charge flow (or current passes) through that point. • Current density at point P is given by dI J n dA ? ? ? + I dA n J I ? ? dA J dA cos ? – • If the cross–sectional area is not normal to the current, but makes an angle ? with the direction of current then dI J dA cos ? ? ? dI = JdA cos ? = J.dA ? ? ? I J . dA ? ? ? ? ? ? ? • Current density J ? is a vector quantity. It's direction is same as that of E ? . It's S.I. unit is ampere/m 2 and dimension [L –2 A]. Example The current density at a point is ? ? ? ? ? ? 4 2 ˆ J 2 10 j Jm . Find the rate of charge flow through a cross sectional area ? ? 2 ˆ ˆ S 2i 3 j cm ? ? ? Solution The rate of flow of charge = current = I = J.dS ? ? ? ? I = J.S ? ? = ? ? 4 2 10 ? ? ? 4 ˆ ˆ ˆ j 2i 3 j 10 A 6A ? ? ? ? ? ? ? ? ? Example A potential difference applied to the ends of a wire made up of an alloy drives a current through it. The current density varies as J = 3 + 2r, where r is the distance of the point from the axis. If R be the radius of the wire, then the total current through any cross section of the wire. Solution Consider a circular strip of radius r and thickness dr dI = J.dS ? ? = ? ? ? ? 3 2r 2 rdr cos0 ? ? ? = ? ? 2 2 3r 2r dr ? ? ? ? R 2 0 I 2 3r 2r dr ? ? ? ? = R 2 3 0 3r 2 2 r 2 3 ? ? ? ? ? ? ? ? = 2 3 3R 2R 2 2 3 ? ? ? ? ? ? ? ? units RELATION BETWEEN CURRENT DENSITY, CONDUCTIVITY AND ELECTRIC FIELD Let the number of free electrons per unit volume in a conductor = n Total number of electrons in dx distance = n (Adx) JEEMAIN.GURU J E E - P h y s i c s E 5 NODE6 (E)\Data\2014\Kota\JEE-Advanced\SMP\Phy\Unit-08\Current electricity\Eng\Theory.p65 Total charge dQ = n (Adx)e Current dQ dx I nAe dt dt ? ? = neAv d , Current density I J A ? = nev d = eE ne m ? ? ? ? ? ? ? ? ? ? ? d eE v m ? ? ? ? ? ? ? ? ? 2 ne J E m ? ? ? ? ? ? ? ? ? J = ?E, where conductivity 2 ne m ? ? ? ? depends only on the material of the conductor and its temperature. In vector form J E ? ? ? ? Ohm's law (at microscopic level) RELATION BETWEEN POTENTIAL DIFFERENCE AND CURRENT (Ohm's Law) If the physical conditions of the conductor (length, temperature, mechanical strain etc.) remains same, then the current flowing through the conductor is directly proportional to the potential difference across it's two ends i.e. I ? V ? V = IR where R is a proportionality constant, known as electric resistance. Ohm's law (at macroscopic level) • Ohm's law is not a universal law. The substances, which obey ohm's law are known as ohmic. • Graph between V and I for a metallic conductor is a straight line as shown. ? ? ? V ? Slope of the line ? ? ? ? V tan R I GOLDEN KEY POINTS • 1 ampere of current means the flow of 6.25 × 10 18 electrons per second through any cross section of conductor. • Current is a scalar quantity but current density is a vector quantity. • Order of free electron density in conductors = 10 28 electrons/m 3 • v T ? ? v d • If a steady current flows in a metallic conductor of non uniform cross section. (i) Along the wire I is same. (ii) Current density and drift velocity depends on area A 1 A 2 I 1 = I 2 , A 1 < A 2 ? J 1 > J 2 , E 1 > E 2 , 1 2 d d v v ? • If the temperature of the conductor increases, the amplitude of the vibrations of the positive ions in the conductor also increase. Due to this, the free electrons collide more frequently with the vibrating ions and as a result, the average relaxation time decreases. • At different temperatures V–I curves are different. Here tan ? 1 > tan ? 2 So R 1 > R 2 i.e. T 1 > T 2 JEEMAIN.GURURead More

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!