Page 1 Introductory Exercise 24.4 Q 1. The current through an inductor of 1H is given by, i = 3t sin t. Find the voltage across the inductor. Solutions 1. Here L = 1H and di dt = 3 [sin t + t cos t] |e| = 3(t cos t+ sin t) Introductory Exercise 24.5 Q 1. (a) Calculate the self inductance of a solenoid that is tightly wound with wire of diameter 0.10 cm, has a cross-sectional area 0.90 cm 2 and is 40 cm long. (b) If the current through the solenoid decreases uniformly from 10 A to 0A in 0.10s, what is the emf induced between the ends of the solenoid? Q 2. An inductor is connected to a battery through a switch. The emf induced in the inductor is much larger when the switch is opened as compared to the emf induced when the switch is closed. Is this statement true or false? Solutions 1. (a) = 4.5 × 10 -5 H (b) = 4.5 × 10 -3 V 2. When switch is opened current suddenly decreasing from steady state value to zero. When switch is closed it takes time to increase from 0 to steady state value. ?t in second case in large. Hence induced emf is less. Introductory Exercise 24.6 Q 1. Two single turn circular loops of wire have radii R and r (R>>r). The loops lie in the same plane and are concentric. Show that the mutual inductance of the pair is 2 0 r 2R ?? . Solutions 1. Magnetic field due to large loop, Area of smaller loop Page 2 Introductory Exercise 24.4 Q 1. The current through an inductor of 1H is given by, i = 3t sin t. Find the voltage across the inductor. Solutions 1. Here L = 1H and di dt = 3 [sin t + t cos t] |e| = 3(t cos t+ sin t) Introductory Exercise 24.5 Q 1. (a) Calculate the self inductance of a solenoid that is tightly wound with wire of diameter 0.10 cm, has a cross-sectional area 0.90 cm 2 and is 40 cm long. (b) If the current through the solenoid decreases uniformly from 10 A to 0A in 0.10s, what is the emf induced between the ends of the solenoid? Q 2. An inductor is connected to a battery through a switch. The emf induced in the inductor is much larger when the switch is opened as compared to the emf induced when the switch is closed. Is this statement true or false? Solutions 1. (a) = 4.5 × 10 -5 H (b) = 4.5 × 10 -3 V 2. When switch is opened current suddenly decreasing from steady state value to zero. When switch is closed it takes time to increase from 0 to steady state value. ?t in second case in large. Hence induced emf is less. Introductory Exercise 24.6 Q 1. Two single turn circular loops of wire have radii R and r (R>>r). The loops lie in the same plane and are concentric. Show that the mutual inductance of the pair is 2 0 r 2R ?? . Solutions 1. Magnetic field due to large loop, Area of smaller loop Introductory Exercise 24.7 Q 1. A sensitive electronic device of resistance 175 ? is to be connected to a source of emf by a switch. The device is designed to operate with a current of 36 mA, but to avoid damage to the device, the current can rise to no more than 4.9 mA in the first 58 ?s after the switch is closed. To protect the device it is connected in series with an inductor. (a) What emf must the source have? (b) What inductance is required? (c) What is the time constant? Q 2. Show that L R has units of time. Q 3. An inductor with an inductance of 3.00 H and a resistance of 7.00 ? is connected to the terminals of a battery with an emf of 12.0 V and negligible internal resistance. Find: (a) The initial rate of increase of current in the circuit. (b) The rate of increase of current at the instant when the current is 1.00 A. (c) The current 0.2 s after the circuit is closed. (d) The final steady state current. Q 4. In the above problem (a) What is the power input to the inductor from the battery as a function of time if the circuit is completed at t = 0? (b) What is the rate of dissipation of energy in the resistance of the inductor as a function of time? (c) What is the rate at which the energy of the magnetic field is increasing, as a function of time? (d) Compare the results of (a), (b) and (c). Q 5. In the simple L-R circuit, can the emf induced across the inductor ever be greater than the emf of the battery used to produce the current? Q 6. The switch in figure is closed at time t = 0. Find the current in the inductor and the current through the switch as functions of time thereafter. Solutions 1. (a) EMF = i 0 R = (36 × 10 -3 ) (175) = 6.3 V (b) Solving we get Page 3 Introductory Exercise 24.4 Q 1. The current through an inductor of 1H is given by, i = 3t sin t. Find the voltage across the inductor. Solutions 1. Here L = 1H and di dt = 3 [sin t + t cos t] |e| = 3(t cos t+ sin t) Introductory Exercise 24.5 Q 1. (a) Calculate the self inductance of a solenoid that is tightly wound with wire of diameter 0.10 cm, has a cross-sectional area 0.90 cm 2 and is 40 cm long. (b) If the current through the solenoid decreases uniformly from 10 A to 0A in 0.10s, what is the emf induced between the ends of the solenoid? Q 2. An inductor is connected to a battery through a switch. The emf induced in the inductor is much larger when the switch is opened as compared to the emf induced when the switch is closed. Is this statement true or false? Solutions 1. (a) = 4.5 × 10 -5 H (b) = 4.5 × 10 -3 V 2. When switch is opened current suddenly decreasing from steady state value to zero. When switch is closed it takes time to increase from 0 to steady state value. ?t in second case in large. Hence induced emf is less. Introductory Exercise 24.6 Q 1. Two single turn circular loops of wire have radii R and r (R>>r). The loops lie in the same plane and are concentric. Show that the mutual inductance of the pair is 2 0 r 2R ?? . Solutions 1. Magnetic field due to large loop, Area of smaller loop Introductory Exercise 24.7 Q 1. A sensitive electronic device of resistance 175 ? is to be connected to a source of emf by a switch. The device is designed to operate with a current of 36 mA, but to avoid damage to the device, the current can rise to no more than 4.9 mA in the first 58 ?s after the switch is closed. To protect the device it is connected in series with an inductor. (a) What emf must the source have? (b) What inductance is required? (c) What is the time constant? Q 2. Show that L R has units of time. Q 3. An inductor with an inductance of 3.00 H and a resistance of 7.00 ? is connected to the terminals of a battery with an emf of 12.0 V and negligible internal resistance. Find: (a) The initial rate of increase of current in the circuit. (b) The rate of increase of current at the instant when the current is 1.00 A. (c) The current 0.2 s after the circuit is closed. (d) The final steady state current. Q 4. In the above problem (a) What is the power input to the inductor from the battery as a function of time if the circuit is completed at t = 0? (b) What is the rate of dissipation of energy in the resistance of the inductor as a function of time? (c) What is the rate at which the energy of the magnetic field is increasing, as a function of time? (d) Compare the results of (a), (b) and (c). Q 5. In the simple L-R circuit, can the emf induced across the inductor ever be greater than the emf of the battery used to produce the current? Q 6. The switch in figure is closed at time t = 0. Find the current in the inductor and the current through the switch as functions of time thereafter. Solutions 1. (a) EMF = i 0 R = (36 × 10 -3 ) (175) = 6.3 V (b) Solving we get L = ( ? L )R = 397 × l72 × 10 -6 = 69 × 10 -3 H = 69 mH 2. has the units of time 3. (a) (b) E = V R + V L = iR + di L dt 12 = (1) (7) + (3) di dt (c) ...(i) t = 0.2 s Substituting the values in Eq. (i), we get the current. (d) Steady state current 4. (a) Power supplied by battery = Ei where E = 12 V and (b) P R = i 2 R (c) (d) Power supplied by battery = P R + P L 5. E = V R + V L 6. At t = 0 inductor offers infinite resistance. Hence current through inductor wire is zero. Whole current passes through two registers of 4 ? each. At t = ?, inductor offers zero resistance. So, main current Page 4 Introductory Exercise 24.4 Q 1. The current through an inductor of 1H is given by, i = 3t sin t. Find the voltage across the inductor. Solutions 1. Here L = 1H and di dt = 3 [sin t + t cos t] |e| = 3(t cos t+ sin t) Introductory Exercise 24.5 Q 1. (a) Calculate the self inductance of a solenoid that is tightly wound with wire of diameter 0.10 cm, has a cross-sectional area 0.90 cm 2 and is 40 cm long. (b) If the current through the solenoid decreases uniformly from 10 A to 0A in 0.10s, what is the emf induced between the ends of the solenoid? Q 2. An inductor is connected to a battery through a switch. The emf induced in the inductor is much larger when the switch is opened as compared to the emf induced when the switch is closed. Is this statement true or false? Solutions 1. (a) = 4.5 × 10 -5 H (b) = 4.5 × 10 -3 V 2. When switch is opened current suddenly decreasing from steady state value to zero. When switch is closed it takes time to increase from 0 to steady state value. ?t in second case in large. Hence induced emf is less. Introductory Exercise 24.6 Q 1. Two single turn circular loops of wire have radii R and r (R>>r). The loops lie in the same plane and are concentric. Show that the mutual inductance of the pair is 2 0 r 2R ?? . Solutions 1. Magnetic field due to large loop, Area of smaller loop Introductory Exercise 24.7 Q 1. A sensitive electronic device of resistance 175 ? is to be connected to a source of emf by a switch. The device is designed to operate with a current of 36 mA, but to avoid damage to the device, the current can rise to no more than 4.9 mA in the first 58 ?s after the switch is closed. To protect the device it is connected in series with an inductor. (a) What emf must the source have? (b) What inductance is required? (c) What is the time constant? Q 2. Show that L R has units of time. Q 3. An inductor with an inductance of 3.00 H and a resistance of 7.00 ? is connected to the terminals of a battery with an emf of 12.0 V and negligible internal resistance. Find: (a) The initial rate of increase of current in the circuit. (b) The rate of increase of current at the instant when the current is 1.00 A. (c) The current 0.2 s after the circuit is closed. (d) The final steady state current. Q 4. In the above problem (a) What is the power input to the inductor from the battery as a function of time if the circuit is completed at t = 0? (b) What is the rate of dissipation of energy in the resistance of the inductor as a function of time? (c) What is the rate at which the energy of the magnetic field is increasing, as a function of time? (d) Compare the results of (a), (b) and (c). Q 5. In the simple L-R circuit, can the emf induced across the inductor ever be greater than the emf of the battery used to produce the current? Q 6. The switch in figure is closed at time t = 0. Find the current in the inductor and the current through the switch as functions of time thereafter. Solutions 1. (a) EMF = i 0 R = (36 × 10 -3 ) (175) = 6.3 V (b) Solving we get L = ( ? L )R = 397 × l72 × 10 -6 = 69 × 10 -3 H = 69 mH 2. has the units of time 3. (a) (b) E = V R + V L = iR + di L dt 12 = (1) (7) + (3) di dt (c) ...(i) t = 0.2 s Substituting the values in Eq. (i), we get the current. (d) Steady state current 4. (a) Power supplied by battery = Ei where E = 12 V and (b) P R = i 2 R (c) (d) Power supplied by battery = P R + P L 5. E = V R + V L 6. At t = 0 inductor offers infinite resistance. Hence current through inductor wire is zero. Whole current passes through two registers of 4 ? each. At t = ?, inductor offers zero resistance. So, main current The distributes in 4 ? and 3 ? in inverse ratio of resistance. Hence current through 4 ? is 1A and through 8 ? is 0.5A. For equivalent ? L of the circuit R net across inductor after short-circuiting the battery is 10 ?. Introductory Exercise 24.8 Q 1. Show that LC has units of time. Q 2. While comparing the L-C oscillations with the oscillations of spring block system, with whom the magnetic energy can be compared and why? Q 3. In an L-C circuit L = 0.75 H and C = 18 ?F. (a) At the instant when the current in the inductor is changing at a rate of 3.40 A/s, what is the charge on the capacitor? (b) When the charge on the capacitor is 4.2 × 10 -4 C, what is the induced emf in the inductor? Q 4. An L-C circuit consists of a 20.0 mH inductor and a 0.5 ?F capacitor. If the maximum instantaneous current is 0.1 A, what is the greatest potential difference across the capacitor? Solutions 1. 2. See the answer 3. (a) V L = V C = (0.75 × 18 × 10 -6 ) (3.4) = 45.9 × 10 -6 C = 45.9 ?C (b) = 23.3 V Page 5 Introductory Exercise 24.4 Q 1. The current through an inductor of 1H is given by, i = 3t sin t. Find the voltage across the inductor. Solutions 1. Here L = 1H and di dt = 3 [sin t + t cos t] |e| = 3(t cos t+ sin t) Introductory Exercise 24.5 Q 1. (a) Calculate the self inductance of a solenoid that is tightly wound with wire of diameter 0.10 cm, has a cross-sectional area 0.90 cm 2 and is 40 cm long. (b) If the current through the solenoid decreases uniformly from 10 A to 0A in 0.10s, what is the emf induced between the ends of the solenoid? Q 2. An inductor is connected to a battery through a switch. The emf induced in the inductor is much larger when the switch is opened as compared to the emf induced when the switch is closed. Is this statement true or false? Solutions 1. (a) = 4.5 × 10 -5 H (b) = 4.5 × 10 -3 V 2. When switch is opened current suddenly decreasing from steady state value to zero. When switch is closed it takes time to increase from 0 to steady state value. ?t in second case in large. Hence induced emf is less. Introductory Exercise 24.6 Q 1. Two single turn circular loops of wire have radii R and r (R>>r). The loops lie in the same plane and are concentric. Show that the mutual inductance of the pair is 2 0 r 2R ?? . Solutions 1. Magnetic field due to large loop, Area of smaller loop Introductory Exercise 24.7 Q 1. A sensitive electronic device of resistance 175 ? is to be connected to a source of emf by a switch. The device is designed to operate with a current of 36 mA, but to avoid damage to the device, the current can rise to no more than 4.9 mA in the first 58 ?s after the switch is closed. To protect the device it is connected in series with an inductor. (a) What emf must the source have? (b) What inductance is required? (c) What is the time constant? Q 2. Show that L R has units of time. Q 3. An inductor with an inductance of 3.00 H and a resistance of 7.00 ? is connected to the terminals of a battery with an emf of 12.0 V and negligible internal resistance. Find: (a) The initial rate of increase of current in the circuit. (b) The rate of increase of current at the instant when the current is 1.00 A. (c) The current 0.2 s after the circuit is closed. (d) The final steady state current. Q 4. In the above problem (a) What is the power input to the inductor from the battery as a function of time if the circuit is completed at t = 0? (b) What is the rate of dissipation of energy in the resistance of the inductor as a function of time? (c) What is the rate at which the energy of the magnetic field is increasing, as a function of time? (d) Compare the results of (a), (b) and (c). Q 5. In the simple L-R circuit, can the emf induced across the inductor ever be greater than the emf of the battery used to produce the current? Q 6. The switch in figure is closed at time t = 0. Find the current in the inductor and the current through the switch as functions of time thereafter. Solutions 1. (a) EMF = i 0 R = (36 × 10 -3 ) (175) = 6.3 V (b) Solving we get L = ( ? L )R = 397 × l72 × 10 -6 = 69 × 10 -3 H = 69 mH 2. has the units of time 3. (a) (b) E = V R + V L = iR + di L dt 12 = (1) (7) + (3) di dt (c) ...(i) t = 0.2 s Substituting the values in Eq. (i), we get the current. (d) Steady state current 4. (a) Power supplied by battery = Ei where E = 12 V and (b) P R = i 2 R (c) (d) Power supplied by battery = P R + P L 5. E = V R + V L 6. At t = 0 inductor offers infinite resistance. Hence current through inductor wire is zero. Whole current passes through two registers of 4 ? each. At t = ?, inductor offers zero resistance. So, main current The distributes in 4 ? and 3 ? in inverse ratio of resistance. Hence current through 4 ? is 1A and through 8 ? is 0.5A. For equivalent ? L of the circuit R net across inductor after short-circuiting the battery is 10 ?. Introductory Exercise 24.8 Q 1. Show that LC has units of time. Q 2. While comparing the L-C oscillations with the oscillations of spring block system, with whom the magnetic energy can be compared and why? Q 3. In an L-C circuit L = 0.75 H and C = 18 ?F. (a) At the instant when the current in the inductor is changing at a rate of 3.40 A/s, what is the charge on the capacitor? (b) When the charge on the capacitor is 4.2 × 10 -4 C, what is the induced emf in the inductor? Q 4. An L-C circuit consists of a 20.0 mH inductor and a 0.5 ?F capacitor. If the maximum instantaneous current is 0.1 A, what is the greatest potential difference across the capacitor? Solutions 1. 2. See the answer 3. (a) V L = V C = (0.75 × 18 × 10 -6 ) (3.4) = 45.9 × 10 -6 C = 45.9 ?C (b) = 23.3 V 4. = 20 V Introductory Exercise 24.9 Q 1. A long solenoid of cross-sectional area 5.0 cm 2 is wound with 25 turns of wire per centimetre. It is placed in the middle of a closely wrapped coil of 10 turns and radius 25 cm as shown. (a) What is the emf induced in the coil when the current through the solenoid is decreasing at a rate -0.20 A/s? (b) What is the electric field induced in the coil? Q 2. For the situation described in figure the magnetic field changes with time according to, B = (2.00 t 3 -4.00 t 2 + 0.8) T and r 2 = 2R = 5.0 cm (a) Calculate the force on an electron located at P 2 at t = 2.00 s (b) What are the magnitude and direction of the electric field at P 1 when t = 3.00 s and r 1 = 0.02 m. Hint: For the direction, see whether the field is increasing or decreasing at given times. Solution 1. In the theory we have already derived mutual inductance between solenoid and coil, (b)Read More

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