A flux of 0.25 mWb is produced by a coil of 1000 turns wound on a ring with a current of 2 A in it. Calculate the e.m.f induced in the coil when a current of 10 A is switched off, assuming the current will fall to zero in 1 millisecond.
This problem involves calculating the electromotive force (e.m.f.) induced in a coil when the current flowing through it changes rapidly. We can solve this using the principles of electromagnetic induction, specifically Faraday's Law of Induction.
Calculate Coil Induced EMF
The induced e.m.f. ($\mathcal{E}$) in a coil is directly proportional to the rate of change of magnetic flux ($\Phi$) through the coil and the number of turns (N) in the coil. This is given by Faraday's Law:
$$ \mathcal{E} = -N \frac{d\Phi}{dt} $$
Alternatively, if the coil has a constant inductance (L), the induced e.m.f. can be calculated using the rate of change of current ($\frac{dI}{dt}$):
$$ \mathcal{E} = -L \frac{dI}{dt} $$
To use the second formula, we first need to determine the inductance (L) of the coil. Inductance is defined as the ratio of the magnetic flux linkage ($N\Phi$) to the current (I) producing it:
$$ L = \frac{N\Phi}{I} $$
Given Parameters
Let's list the values provided in the question:
| Parameter | Symbol | Value |
|---|---|---|
| Magnetic Flux | $\Phi$ | 0.25 mWb = $0.25 \times 10^{-3}$ Wb |
| Number of Turns | N | 1000 |
| Initial Current (for flux generation) | I | 2 A |
| Current Switched Off (Initial for change) | $I_{initial}$ | 10 A |
| Final Current (after switching off) | $I_{final}$ | 0 A |
| Time for Current Change | $\Delta t$ | 1 ms = $1 \times 10^{-3}$ s |
Note: The flux value (0.25 mWb) is produced by a 2 A current, which helps us find the coil's inductance. The e.m.f calculation itself depends on the change from 10 A to 0 A.
Calculate Coil Inductance
Using the initial condition where a current of 2 A produces a flux of 0.25 mWb:
$$ L = \frac{N\Phi}{I} = \frac{1000 \times (0.25 \times 10^{-3} \text{ Wb})}{2 \text{ A}} $$
$$ L = \frac{0.25 \text{ Wb}}{2 \text{ A}} = 0.125 \text{ H} $$
So, the inductance of the coil is 0.125 Henry (H).
Calculate Induced EMF
Now, we calculate the induced e.m.f. when the current changes from 10 A to 0 A in 1 millisecond.
First, find the change in current ($\Delta I$):
$$ \Delta I = I_{final} - I_{initial} = 0 \text{ A} - 10 \text{ A} = -10 \text{ A} $$
The time interval ($\Delta t$) is given as 1 ms or $1 \times 10^{-3}$ s.
The rate of change of current ($\frac{dI}{dt}$) can be approximated as $\frac{\Delta I}{\Delta t}$:
$$ \frac{dI}{dt} = \frac{-10 \text{ A}}{1 \times 10^{-3} \text{ s}} = -10000 \text{ A/s} $$
Now, substitute the values of L and $\frac{dI}{dt}$ into the e.m.f. formula:
$$ \mathcal{E} = -L \frac{dI}{dt} = -(0.125 \text{ H}) \times (-10000 \text{ A/s}) $$
$$ \mathcal{E} = 1250 \text{ V} $$
Therefore, the e.m.f. induced in the coil is 1250 Volts.
Important Questions from Faraday's Law
In the table shown, List I and List II, respectively, contain terms appearing on the left-hand side and the right-hand side of Maxwell’s equations (in their standard form). Match the left-hand side with the corresponding right-hand side.
List I
List II
1.
∇. D
P
0
2.
∇ × E
Q
\(\rho_v\)
3.
∇. B
R
\( - \frac{{\partial B}}{{\partial t}}\)
4.
∇ × H
S
\(J+ \frac{{\partial D}}{{\partial t}}\)
“By the motion of the conductor or the coil in a magnetic field, i.e., the magnetic field is stationary and the moving conductors cut through it. The EMF generated in this way is normally called dynamically induced EMF.”
The given statement is specified by which of the following laws?
Which of the following law states that “whenever the magnetic flux linked with a conductor or coil changes, an emf is induced in it?
All induction type transducers are based on
The figure shows an arbitrarily shaped planar conducting loop A in the XY plane. Two nonintersecting regions with areas $a_1$ and $a_2$ within the loop are subjected to magnetic fields $\vec{B}_1 = \frac{m}{\sqrt{2}} \sin(\omega t) (1\hat{x} + 0\hat{y} + 1\hat{z})$, and
$\vec{B}_2 = -\frac{n}{\sqrt{2}} \cos(2\omega t + \pi/4) (0\hat{x} + 1\hat{y} + 1\hat{z})$, respectively.
What is the expression for the induced rms voltage in loop A?
