A small square loop of edge a and resistance R is moved with velocity `v_(0)` away from an infinitely long current carrying conductor carrying current I so that the conductor and side of square are always in same plane. Find induced current in loop at a separation of `r`

Options

A. `(mu_(0)ia^(2)v)/(pi r^(2)R)`
B. `(mu_(0)ia^(2)v)/(4pi r^(2)R)`
C. `(mu_(0)ia^(2)v)/(2pi r^(2)R)`
D. `(mu_(0)iav)/(2pi rR)`

Explanation

To find the induced current in a small square loop of edge \( a \) and resistance \( R \) that is moved with velocity \( v_0 \) away from an infinitely long current-carrying conductor carrying current \( I \), we can follow these steps:

### Step 1: Determine the Magnetic Field
The magnetic field \( B \) at a distance \( r \) from an infinitely long straight conductor carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Express the Distance
As the loop moves away from the conductor with velocity \( v_0 \), the distance \( r \) from the conductor at time \( t \) can be expressed as:
\[ r = r_0 + v_0 t \]
where \( r_0 \) is the initial distance from the conductor.

### Step 3: Calculate the Magnetic Field as a Function of Time
Substituting \( r \) into the magnetic field equation, we get:
\[ B(t) = \frac{\mu_0 I}{2 \pi (r_0 + v_0 t)} \]

### Step 4: Calculate the Magnetic Flux
The magnetic flux \( \Phi \) through the square loop of area \( A = a^2 \) is given by:
\[ \Phi = B(t) \cdot A = \frac{\mu_0 I}{2 \pi (r_0 + v_0 t)} \cdot a^2 \]

### Step 5: Differentiate the Magnetic Flux to Find EMF
The induced EMF \( \mathcal{E} \) is given by Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Differentiating the flux with respect to time:
\[ \mathcal{E} = -\frac{d}{dt} \left( \frac{\mu_0 I a^2}{2 \pi (r_0 + v_0 t)} \right) \]
Using the chain rule, we find:
\[ \mathcal{E} = \frac{\mu_0 I a^2 v_0}{2 \pi (r_0 + v_0 t)^2} \]

### Step 6: Calculate the Induced Current
Using Ohm's law, the induced current \( I' \) can be calculated as:
\[ I' = \frac{\mathcal{E}}{R} = \frac{\mu_0 I a^2 v_0}{2 \pi R (r_0 + v_0 t)^2} \]

### Final Result
Thus, the induced current in the loop at a separation of \( r \) is:
\[ I' = \frac{\mu_0 I a^2 v_0}{2 \pi R (r_0 + v_0 t)^2} \]

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