At a place earth's magnetic field, `5xx10^(5)` `Wb//m^(2)` is acting perpendicular to a coil of radius R=5cm . If `mu_(0)//4pi=10^(-7)` , then how much current is induced in circular loop ?

Options

A. 0.2 A
B. `0 A`
C. `4 A`
D. 40 A

Explanation

To solve the problem, we need to determine the induced current in a circular loop placed in a magnetic field. Here’s a step-by-step solution:

### Step 1: Understand the Given Data
- Magnetic field (B) = \(5 \times 10^{-5} \, \text{Wb/m}^2\)
- Radius of the coil (R) = 5 cm = 0.05 m
- \(\mu_0 / 4\pi = 10^{-7}\)

### Step 2: Calculate the Area of the Circular Coil
The area (A) of a circular coil is given by the formula:
\[ A = \pi R^2 \]
Substituting the value of R:
\[ A = \pi (0.05)^2 = \pi (0.0025) \approx 0.00785 \, \text{m}^2 \]

### Step 3: Calculate the Magnetic Flux (Φ)
Magnetic flux (Φ) through the coil is given by:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
Since the magnetic field is acting perpendicular to the coil, \(\theta = 0\) degrees, and \(\cos(0) = 1\):
\[ \Phi = B \cdot A = (5 \times 10^{-5}) \cdot (0.00785) \approx 3.925 \times 10^{-7} \, \text{Wb} \]

### Step 4: Determine the Change in Magnetic Flux
Since the problem states that the magnetic field is constant and does not change with time, the change in magnetic flux (\(d\Phi/dt\)) is zero:
\[ \frac{d\Phi}{dt} = 0 \]

### Step 5: Calculate the Induced EMF (E)
According to Faraday's law of electromagnetic induction, the induced EMF (E) is given by:
\[ E = -\frac{d\Phi}{dt} \]
Since \(\frac{d\Phi}{dt} = 0\):
\[ E = 0 \]

### Step 6: Calculate the Induced Current (I)
Using Ohm's law, the induced current (I) can be calculated as:
\[ I = \frac{E}{R} \]
Where R is the resistance of the coil. However, since \(E = 0\):
\[ I = 0 \]

### Conclusion
The induced current in the circular loop is \(0 \, \text{A}\).

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