A circular loop of radius `1m` is placed in a varying magnetic field given as `B=6t` Tesla, where `t` is time in sec.
(a)Find the `emf` induced in the coil if the plane of the coil is perpendicular to the magnetic field.
(b) Find the electric field in the tangential directin, induced due to the changing magnetic field.
(c)Find the current in the loop if its resistance is `1Omega//m`.

Explanation

Let's solve the problem step by step.

### Given:
- Radius of the circular loop, \( r = 1 \, \text{m} \)
- Magnetic field, \( B(t) = 6t \, \text{T} \) (where \( t \) is time in seconds)
- Resistance of the loop, \( R = 1 \, \Omega/\text{m} \)

### (a) Find the emf induced in the coil if the plane of the coil is perpendicular to the magnetic field.

**Step 1:** Calculate the area of the circular loop.
\[ A = \pi r^2 = \pi (1)^2 = \pi \, \text{m}^2 \]

**Step 2:** Find the rate of change of the magnetic field, \( \frac{dB}{dt} \).
\[ \frac{dB}{dt} = \frac{d(6t)}{dt} = 6 \, \text{T/s} \]

**Step 3:** Use Faraday's law of electromagnetic induction to find the induced emf (\( \mathcal{E} \)).
\[ \mathcal{E} = -A \frac{dB}{dt} = -\pi \cdot 6 = -6\pi \, \text{V} \]
(Note: The negative sign indicates the direction of induced emf according to Lenz's law, but we can consider the magnitude for this problem.)

**Result:**
\[ \mathcal{E} = 6\pi \, \text{V} \]

### (b) Find the electric field in the tangential direction, induced due to the changing magnetic field.

**Step 1:** Use the relationship between the induced emf and the electric field.
\[ \mathcal{E} = E \cdot 2\pi r \]

**Step 2:** Substitute the values to find the electric field \( E \).
\[ E \cdot 2\pi(1) = 6\pi \]
\[ E = \frac{6\pi}{2\pi} = 3 \, \text{V/m} \]

**Result:**
\[ E = 3 \, \text{V/m} \]

### (c) Find the current in the loop if its resistance is \( 1 \, \Omega/\text{m} \).

**Step 1:** Calculate the total resistance of the loop.
Since the resistance is given per meter and the circumference of the loop is \( 2\pi r \):
\[ R = 1 \, \Omega/\text{m} \times 2\pi(1) = 2\pi \, \Omega \]

**Step 2:** Use Ohm's law to find the current \( I \).
\[ I = \frac{\mathcal{E}}{R} = \frac{6\pi}{2\pi} = 3 \, \text{A} \]

**Result:**
\[ I = 3 \, \text{A} \]

### Summary of Results:
- (a) Induced emf, \( \mathcal{E} = 6\pi \, \text{V} \)
- (b) Electric field, \( E = 3 \, \text{V/m} \)
- (c) Current, \( I = 3 \, \text{A} \)