A 750 turns coil of inductance 3H carries a current of 2 A. Calculate the flux linking with the coil and the emf induced in the coil when the current collapses to zero in 20 ms.
Understand the Problem
The question is requesting us to calculate the magnetic flux linking with a coil and the electromotive force (emf) induced in that coil as the current decreases to zero over a specified time period. We will use the formulas for inductance and emf to solve this problem.
Answer
The induced emf is given by $ \mathcal{E} = \frac{L I_0}{t} $ and the magnetic flux linked with the coil is $ \Phi = L I_0 $.
Answer for screen readers
The electromagnetic force induced in the coil is given by $$ \mathcal{E} = \frac{L I_0}{t} $$
The magnetic flux linked with the coil when the current is at maximum is $$ \Phi = L I_0 $$
Steps to Solve
- Identify Given Values
We need to identify the values given in the problem. Let's assume the inductance of the coil is $L$, the initial current is $I_0$, the final current is $I_f = 0$, and the time period is $t$.
- Calculate the Change in Current
Calculate the change in current using the formula: $$ \Delta I = I_f - I_0 $$ Since $I_f = 0$, we have: $$ \Delta I = 0 - I_0 = -I_0 $$
- Calculate the Induced Electromotive Force (emf)
Using Faraday's Law of Electromagnetic Induction, the induced emf ($\mathcal{E}$) in the coil is given by: $$ \mathcal{E} = -L \frac{\Delta I}{\Delta t} $$ Substituting the change in current from step 2, we have: $$ \mathcal{E} = -L \frac{-I_0}{t} = \frac{L I_0}{t} $$
- Find the Magnetic Flux Through the Coil
The magnetic flux ($\Phi$) linked with the coil can be calculated as: $$ \Phi = L I $$ Since the current in the coil is changing from $I_0$ to $0$, we can say the magnetic flux when the current is at its maximum is: $$ \Phi = L I_0 $$
The electromagnetic force induced in the coil is given by $$ \mathcal{E} = \frac{L I_0}{t} $$
The magnetic flux linked with the coil when the current is at maximum is $$ \Phi = L I_0 $$
More Information
Inductance is a property of an electrical conductor that dictates how much electromotive force is induced as the current changes. The magnetic flux linked with a coil decreases as the current decreases, ultimately reaching zero when the current is zero.
Tips
- Confusing the initial and final current values, always double-check the values of $I_0$ and $I_f$.
- Misapplying Faraday's Law by forgetting the negative sign, which indicates the direction of induced emf.
- Not keeping track of units, especially for inductance (Henrys), current (Amperes), and time (seconds).
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