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Question: In \(0.1\;s\), the current in a coil increases from \(1\;A\) to \(1.5\;A\). If the inductance of coi...

In 0.1  s0.1\;s, the current in a coil increases from 1  A1\;A to 1.5  A1.5\;A. If the inductance of coil is 60  mH60\;mH, then what will be the induced current in external resistance of 3  Ω3\;\Omega ?
A. 1  A1\;A
B. 0.5  A0.5\;A
C. 0.2  A0.2\;A
D. 0.1  A0.1\;A

Explanation

Solution

Hint: In the question, the current flows through the coil increases with a definite time and the inductance of the coil is given. By using the induced emf formula for the inductance coil, the value of the emf induced in the coil is calculated. Then using the Ohm’s law, the value of the induced current will be calculated.

Useful formula:
The induced emf in the inductance coil is given by,
Vind=Ldidt{V_{ind}} = L\dfrac{{di}}{{dt}}
Where, Vind{V_{ind}} is the emf induced in the coil, LL is the inductance of the coil, didt\dfrac{{di}}{{dt}} is the change in the current with respect to time.

Ohm’s law is given by,
V=IRV = IR
Where, VV is the emf of the circuit, II is the current flows through the circuit and RR is the resistance.

Given data:
The inductance of coil, L=60  mHL = 60\;mH
The current in a coil increases from 1  A1\;A to 1.5  A1.5\;A
The time taken, t=0.1  st = 0.1\;s
External resistance, R=3  ΩR = 3\;\Omega

Complete step by step solution:
The emf induced in the coil is given by,
Vind=Ldidt  ................................................(1){V_{ind}} = L\dfrac{{di}}{{dt}}\;................................................\left( 1 \right)
Where, didt\dfrac{{di}}{{dt}} = (Change in current)/Time
Substitute the given values in equation (1), we get
Vind=60  mH×(1.5  A−1  A)0.1  s   Vind=60  mH×0.5  A0.1  s   Vind=300  mV  {V_{ind}} = 60\;mH \times \dfrac{{\left( {1.5\;A - 1\;A} \right)}}{{0.1\;s}}\; \\\ {V_{ind}} = 60\;mH \times \dfrac{{0.5\;A}}{{0.1\;s}}\; \\\ {V_{ind}} = 300\;mV \\\

By Ohm’s law, we get
Vind=IindR{V_{ind}} = {I_{ind}}R
Where, Vind{V_{ind}} is the emf induced, Iind{I_{ind}} is the current induced and RR is the resistance.
Substitute the given values in Ohm’s law equation, we get
300  mV=Iind×3  Ω Iind=300  mV3  Ω Iind=100  mA Iind=0.1  A  300\;mV = {I_{ind}} \times 3\;\Omega \\\ {I_{ind}} = \dfrac{{300\;mV}}{{3\;\Omega }} \\\ {I_{ind}} = 100\;mA \\\ {I_{ind}} = 0.1\;A \\\

Hence, the option (D) is correct.

Note: From the question, the inductance coil and the resistor of given resistance are connected in a circuit. The emf induced in the inductance coil will flow through the resistor, that induced current in the resistor. Thus, the relation given by Ohm's law is used to calculate the induced current in the external resistance of the circuit which is given.