The acceleration of the charge on the gap faces will cease when the total electric field within the ring becomes zero. For this to happen, the electric field `E_(0)` in the gap is

Options

A. (a) `E_(0) = (abeta)/(delta)`
B. (b) `E_(0) = (2a beta)/(delta)`
C. ( c) `E_(0)` is dependent of `R` for `R gt sqrt((a)/( pi))`
D. ( d) `E_(0)` is independent of `R` for `R gt sqrt((a)/( pi))`

Explanation

To solve the problem, we need to determine the electric field \( E_0 \) in the gap when the total electric field within the ring becomes zero. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
The problem states that the acceleration of a charge in the gap will cease when the electric field in the gap becomes zero. We need to find the expression for the electric field \( E_0 \) in terms of the given parameters.

### Step 2: Identify Given Information
We are given:
- The magnetic field \( B(t) = B_0 + \beta t \), where \( \beta > 0 \).
- We need to find the electric field \( E_0 \) when the electric field in the gap is zero.

### Step 3: Calculate Magnetic Flux
The magnetic flux \( \Phi \) through the area \( A \) of the solenoid is given by:
\[ \Phi = B \cdot A = (B_0 + \beta t) \cdot A \]

### Step 4: Determine the Induced EMF
The induced electromotive force (EMF) \( \mathcal{E} \) can be calculated using Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Differentiating the magnetic flux:
\[ \mathcal{E} = -\frac{d}{dt}[(B_0 + \beta t) \cdot A] = -\left(\beta A\right) \]
Thus, the magnitude of the induced EMF is:
\[ |\mathcal{E}| = \beta A \]

### Step 5: Relate EMF to Electric Field
The relationship between the induced EMF and the electric field \( E \) across the gap of width \( \delta \) is given by:
\[ \mathcal{E} = E \cdot \delta \]
Substituting the value of EMF:
\[ \beta A = E \cdot \delta \]

### Step 6: Solve for Electric Field \( E \)
Rearranging the equation to solve for \( E \):
\[ E = \frac{\beta A}{\delta} \]
This \( E \) is the electric field in the gap.

### Step 7: Final Expression for \( E_0 \)
Since we are looking for \( E_0 \) when the electric field in the gap becomes zero, we can conclude:
\[ E_0 = \frac{\beta A}{\delta} \]

### Step 8: Conditions for Independence of \( R \)
The problem also states that \( E_0 \) is independent of \( R \) if the area of the solenoid \( A \) is significantly smaller than the area of the ring. This leads to the condition:
\[ R^2 > \frac{A}{\pi} \]

### Conclusion
Thus, the electric field \( E_0 \) in the gap is given by:
\[ E_0 = \frac{\beta A}{\delta} \]
And it is independent of \( R \) under the condition that \( R^2 > \frac{A}{\pi} \).

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