Winding wire with insulated coating is used to make a circular ring whose area is A. This ring is kept perpendicular to a uniform magnetic field B. If ring is suddenly squeezed to zero area then how much charge will flow through the wire? R is resistance of winding wire.

Options

A. Zero
B. `(BA)//(2R)`
C. `2(BA)//(R)`
D. `(BA)//(R)`

Explanation

To solve the problem, we need to determine the amount of charge that flows through the wire when the area of the circular ring is suddenly reduced to zero in the presence of a uniform magnetic field. We will follow these steps:

### Step 1: Calculate the Initial Magnetic Flux
The magnetic flux (Φ) through the ring is given by the formula:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
Since the ring is perpendicular to the magnetic field, the angle \( \theta = 0^\circ \) and \( \cos(0) = 1 \). Therefore, the initial magnetic flux is:
\[ \Phi_1 = B \cdot A \]

### Step 2: Calculate the Change in Magnetic Flux
When the area of the ring is suddenly squeezed to zero, the final magnetic flux (Φ₂) becomes:
\[ \Phi_2 = B \cdot 0 = 0 \]
The change in magnetic flux (ΔΦ) is then:
\[ \Delta \Phi = \Phi_2 - \Phi_1 = 0 - (B \cdot A) = -B \cdot A \]

### Step 3: Calculate the Induced EMF
According to Faraday's law of electromagnetic induction, the induced EMF (E) in the loop is given by the rate of change of magnetic flux:
\[ E = -\frac{\Delta \Phi}{\Delta t} \]
Substituting the change in flux:
\[ E = -\frac{-B \cdot A}{\Delta t} = \frac{B \cdot A}{\Delta t} \]

### Step 4: Calculate the Current in the Wire
Using Ohm's law, the current (I) flowing through the wire can be calculated as:
\[ I = \frac{E}{R} \]
Substituting the expression for induced EMF:
\[ I = \frac{B \cdot A}{R \cdot \Delta t} \]

### Step 5: Calculate the Charge Flowing Through the Wire
The charge (ΔQ) that flows through the wire can be calculated using the relationship between current and charge:
\[ I = \frac{\Delta Q}{\Delta t} \]
Rearranging gives:
\[ \Delta Q = I \cdot \Delta t \]
Substituting the expression for current:
\[ \Delta Q = \left(\frac{B \cdot A}{R \cdot \Delta t}\right) \cdot \Delta t \]
The Δt cancels out:
\[ \Delta Q = \frac{B \cdot A}{R} \]

### Final Answer
Thus, the charge that flows through the wire when the area is suddenly squeezed to zero is:
\[ \Delta Q = \frac{B \cdot A}{R} \]