Lecture 11 - Capacitance - Electrostatics Notes | EduRev

: Lecture 11 - Capacitance - Electrostatics Notes | EduRev

 Page 1


Module 2 : Electrostatics
Lecture 11 : Capacitance
 
Objectives
 In this lecture you will learn the following
Capacitors in series and in parallel
Properties of dielectric
Conductor and dielectric in an electric field.
Polarization and bound charges
Gauss's Law for dielectrics
Capacitors in Combination :
Capacitors can be combined in series or parallel combinations in a circuit.
Parallel Combination
When they are in parallel, the potential difference across each capacitor is the same.
The charge on each capacitor is obtained by multiplying with the capacitance, i.e. 
.
Page 2


Module 2 : Electrostatics
Lecture 11 : Capacitance
 
Objectives
 In this lecture you will learn the following
Capacitors in series and in parallel
Properties of dielectric
Conductor and dielectric in an electric field.
Polarization and bound charges
Gauss's Law for dielectrics
Capacitors in Combination :
Capacitors can be combined in series or parallel combinations in a circuit.
Parallel Combination
When they are in parallel, the potential difference across each capacitor is the same.
The charge on each capacitor is obtained by multiplying with the capacitance, i.e. 
.
Since total charge in the capacitors is sum of all the charges, the effective capacitance of the combination is
              
Series Combination :
When capacitors are joined end to end in series, the first capacitor gets charged and induces an equal charge on
the second capacitor which is connected to it. This in turn induces an equal charge on the third capacitor, and
so on.
                                              
The net potential difference between the positive plate of the first capacitor and the negative plate of the last
capacitor in series is
The individual voltage drops are
so that
The effective capacitance is, therefore, given by
Example 19
Calculate the voltage across the 5 F capacitor in the following circuit.
Page 3


Module 2 : Electrostatics
Lecture 11 : Capacitance
 
Objectives
 In this lecture you will learn the following
Capacitors in series and in parallel
Properties of dielectric
Conductor and dielectric in an electric field.
Polarization and bound charges
Gauss's Law for dielectrics
Capacitors in Combination :
Capacitors can be combined in series or parallel combinations in a circuit.
Parallel Combination
When they are in parallel, the potential difference across each capacitor is the same.
The charge on each capacitor is obtained by multiplying with the capacitance, i.e. 
.
Since total charge in the capacitors is sum of all the charges, the effective capacitance of the combination is
              
Series Combination :
When capacitors are joined end to end in series, the first capacitor gets charged and induces an equal charge on
the second capacitor which is connected to it. This in turn induces an equal charge on the third capacitor, and
so on.
                                              
The net potential difference between the positive plate of the first capacitor and the negative plate of the last
capacitor in series is
The individual voltage drops are
so that
The effective capacitance is, therefore, given by
Example 19
Calculate the voltage across the 5 F capacitor in the following circuit.
Solution : 
The equivalent circuit is shown above.The two 10  capacitors in series is equivalent to a 5 
capacitor.5  in parallel with this equivalent capacitor gives 10  as the next equivalent.The circuit
therefore consists of a 10  in series with the 20  capacitor. Since charge remains constant in a series
combination, the potential drop across the 10  capacitor is twice as much as that across 20 
capacitor. The voltage drop across the 10 (and hence across the given 5 ) is V.
                                           
Exercise 2
Determine the effective capacitance of the following capacitance circuit and find the voltage across each
capacitance if the voltage across the points a and b is 300 V.
Page 4


Module 2 : Electrostatics
Lecture 11 : Capacitance
 
Objectives
 In this lecture you will learn the following
Capacitors in series and in parallel
Properties of dielectric
Conductor and dielectric in an electric field.
Polarization and bound charges
Gauss's Law for dielectrics
Capacitors in Combination :
Capacitors can be combined in series or parallel combinations in a circuit.
Parallel Combination
When they are in parallel, the potential difference across each capacitor is the same.
The charge on each capacitor is obtained by multiplying with the capacitance, i.e. 
.
Since total charge in the capacitors is sum of all the charges, the effective capacitance of the combination is
              
Series Combination :
When capacitors are joined end to end in series, the first capacitor gets charged and induces an equal charge on
the second capacitor which is connected to it. This in turn induces an equal charge on the third capacitor, and
so on.
                                              
The net potential difference between the positive plate of the first capacitor and the negative plate of the last
capacitor in series is
The individual voltage drops are
so that
The effective capacitance is, therefore, given by
Example 19
Calculate the voltage across the 5 F capacitor in the following circuit.
Solution : 
The equivalent circuit is shown above.The two 10  capacitors in series is equivalent to a 5 
capacitor.5  in parallel with this equivalent capacitor gives 10  as the next equivalent.The circuit
therefore consists of a 10  in series with the 20  capacitor. Since charge remains constant in a series
combination, the potential drop across the 10  capacitor is twice as much as that across 20 
capacitor. The voltage drop across the 10 (and hence across the given 5 ) is V.
                                           
Exercise 2
Determine the effective capacitance of the following capacitance circuit and find the voltage across each
capacitance if the voltage across the points a and b is 300 V.
                                  
   [Ans. 8 F., 100V,200V,200V,200V,200V,100V]
Conductors and Dielectric
 A conductor is characterized by existence of free electrons . These are electrons in the outermost shells of
atoms (the valence electrons) which get detatched from the parent atoms during the formation of metallic
bonds and move freely in the entire medium in such way that the conductor becomes an equipotential volume. 
In contrast, in dielectrics (insulators), the outer electrons remain bound to the atoms or molecules to which
they belong. Both conductors and dielectric, on the whole, are charge neutral. However, in case of dielectrics,
the charge neutrality is satisfied over much smaller regions (e.g. at molecular level).
Polar and non-polar molecules :
A dielectric consists of molecules which remain locally charge neutral. The molecules may be polar or non-polar.
In non-polar molecules, the charge centres of positive and negative charges coincide so that the net dipole
moment of each molecule is zero. Carbon dioxide molecule is an example of a non-polar molecule.
           
Click here for Animation
Page 5


Module 2 : Electrostatics
Lecture 11 : Capacitance
 
Objectives
 In this lecture you will learn the following
Capacitors in series and in parallel
Properties of dielectric
Conductor and dielectric in an electric field.
Polarization and bound charges
Gauss's Law for dielectrics
Capacitors in Combination :
Capacitors can be combined in series or parallel combinations in a circuit.
Parallel Combination
When they are in parallel, the potential difference across each capacitor is the same.
The charge on each capacitor is obtained by multiplying with the capacitance, i.e. 
.
Since total charge in the capacitors is sum of all the charges, the effective capacitance of the combination is
              
Series Combination :
When capacitors are joined end to end in series, the first capacitor gets charged and induces an equal charge on
the second capacitor which is connected to it. This in turn induces an equal charge on the third capacitor, and
so on.
                                              
The net potential difference between the positive plate of the first capacitor and the negative plate of the last
capacitor in series is
The individual voltage drops are
so that
The effective capacitance is, therefore, given by
Example 19
Calculate the voltage across the 5 F capacitor in the following circuit.
Solution : 
The equivalent circuit is shown above.The two 10  capacitors in series is equivalent to a 5 
capacitor.5  in parallel with this equivalent capacitor gives 10  as the next equivalent.The circuit
therefore consists of a 10  in series with the 20  capacitor. Since charge remains constant in a series
combination, the potential drop across the 10  capacitor is twice as much as that across 20 
capacitor. The voltage drop across the 10 (and hence across the given 5 ) is V.
                                           
Exercise 2
Determine the effective capacitance of the following capacitance circuit and find the voltage across each
capacitance if the voltage across the points a and b is 300 V.
                                  
   [Ans. 8 F., 100V,200V,200V,200V,200V,100V]
Conductors and Dielectric
 A conductor is characterized by existence of free electrons . These are electrons in the outermost shells of
atoms (the valence electrons) which get detatched from the parent atoms during the formation of metallic
bonds and move freely in the entire medium in such way that the conductor becomes an equipotential volume. 
In contrast, in dielectrics (insulators), the outer electrons remain bound to the atoms or molecules to which
they belong. Both conductors and dielectric, on the whole, are charge neutral. However, in case of dielectrics,
the charge neutrality is satisfied over much smaller regions (e.g. at molecular level).
Polar and non-polar molecules :
A dielectric consists of molecules which remain locally charge neutral. The molecules may be polar or non-polar.
In non-polar molecules, the charge centres of positive and negative charges coincide so that the net dipole
moment of each molecule is zero. Carbon dioxide molecule is an example of a non-polar molecule.
           
Click here for Animation
                                                        
        Click here for Animation
In a polar molecules, the arrangement of atoms is such that the molecule has a permanent dipole moment
because of charge separation. Water molecule is an example of a polar molecule.
When a non-polar molecule is put in an electric field, the electric forces cause a small separation of the charges.
The molecule thereby acquires an induced dipole moment. 
A polar molecule, which has a dipole moment in the absence of the electric field, gets its dipole moment aligned
in the direction of the field. In addition, the magnitude of the dipole moment may also increase because of
increased separation of the charges. 
Click here for Animation
Conductor in an Electric Field
Consider what happens when a conductor is placed in an electric field, say, between the plates of a parallel
plate capacitor.
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