How does the electric field (E) between the plates of a charged cylindrical capacitor vary with the distance r from the axis of the cylinder ?

Options

A. `Eprop1/r^2`
B. `Eprop1/r`
C. `Epropr^2`
D. `Epropr`

Answer

To determine how the electric field (E) between the plates of a charged cylindrical capacitor varies with the distance (r) from the axis of the cylinder, we can follow these steps:

### Step 1: Understand the Configuration
We have a cylindrical capacitor consisting of two coaxial cylinders. The inner cylinder has a linear charge density of +λ, and the outer cylinder has a linear charge density of -λ. The outer cylinder is grounded, meaning its potential is zero.

### Step 2: Identify the Regions
1. Inside the inner cylinder (r < B): The electric field (E) is zero because there is no charge enclosed.
2. Between the inner and outer cylinders (B < r < A): This is the region where we need to determine the electric field.
3. Outside the outer cylinder (r > A): The electric field (E) is again zero.

### Step 3: Apply Gauss's Law
To find the electric field in the region between the cylinders (B < r < A), we apply Gauss's Law, which states:

\[ \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]

### Step 4: Choose a Gaussian Surface
We consider a cylindrical Gaussian surface of radius r (where B < r < A) and height L. The area vector \( d\mathbf{A} \) is directed outward, and the electric field \( \mathbf{E} \) is also directed outward.

### Step 5: Calculate the Left Side of Gauss's Law
The left side of Gauss's Law becomes:

\[ \oint \mathbf{E} \cdot d\mathbf{A} = E \cdot (2\pi r L) \]

### Step 6: Calculate the Charge Enclosed
The charge enclosed by the Gaussian surface is given by the linear charge density of the inner cylinder:

\[ Q_{\text{enc}} = \lambda L \]

### Step 7: Set Up the Equation
Substituting the left and right sides into Gauss's Law gives:

\[ E \cdot (2\pi r L) = \frac{\lambda L}{\epsilon_0} \]

### Step 8: Solve for Electric Field (E)
Now, we can solve for the electric field \( E \):

\[ E = \frac{\lambda}{2\pi r \epsilon_0} \]

### Step 9: Analyze the Result
From the equation \( E = \frac{\lambda}{2\pi r \epsilon_0} \), we see that the electric field \( E \) is inversely proportional to the distance \( r \) from the axis of the cylinder. This means that as you move away from the inner cylinder, the electric field decreases.

### Conclusion
The electric field between the plates of a charged cylindrical capacitor varies inversely with the distance \( r \) from the axis of the cylinder.

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