Question

A coil of resistance $400\Omega$ is placed in magnetic field. If the magnetic flux ' $\phi$ ( Wb ) linked with the coil varies with time ' $t$ ( s ) is $\phi = 50t^2 + 4$, the current in the coil at $t = 2\text{ s}$ will be}

• A. 1 A
• B. 2 A
• C. 0.5 A
• D. 0.1 A

Hint:

In induction problems: \[ \mathcal{E}=\left|\frac{d\phi}{dt}\right| \] Differentiate first, then divide by resistance to get current.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
According to Faraday's Law, a changing magnetic flux induces an electromotive force (EMF) in the coil. This EMF then drives an induced current according to Ohm's Law.
Step 2: Key Formula or Approach:
1. Faraday's Law: $e = \left| \frac{d\phi}{dt} \right|$
2. Ohm's Law: $I = \frac{e}{R}$
Step 3: Detailed Explanation:
Given $\phi = 50t^2 + 4$ and $R = 400\Omega$.
1. Find induced EMF by differentiating flux with respect to time:
\[ e = \frac{d}{dt}(50t^2 + 4) = 100t \]
2. Find EMF at $t = 2\text{ s}$:
\[ e = 100 \times 2 = 200\text{ V} \]
3. Calculate current:
\[ I = \frac{e}{R} = \frac{200}{400} = 0.5\text{ A} \]
Step 4: Final Answer:
The current is $0.5\text{ A}$.