The maximum electric field at a point on the axis of a uniformly charged ring is `E_(0)`. At how many points on the axis will the magnitude of the electric field be `E_(0)//2`.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem of finding the number of points on the axis of a uniformly charged ring where the electric field is equal to \( \frac{E_0}{2} \), we can follow these steps:

### Step 1: Understand the Electric Field of a Charged Ring
The electric field \( E \) along the axis of a uniformly charged ring varies with the distance \( x \) from the center of the ring. The electric field reaches its maximum value \( E_0 \) at a certain point along the axis.

### Step 2: Identify the Maximum Electric Field
From the problem statement, we know that the maximum electric field at a point on the axis of the ring is \( E_0 \). This maximum occurs at a specific distance from the center of the ring.

### Step 3: Analyze the Electric Field Graph
The electric field \( E \) as a function of distance \( x \) from the center of the ring can be represented graphically. The graph will show that the electric field starts at zero, increases to a maximum \( E_0 \), and then decreases back to zero as you move further away from the ring.

### Step 4: Determine Points Where \( E = \frac{E_0}{2} \)
To find the points where the electric field is \( \frac{E_0}{2} \), we look at the graph of \( E \) versus \( x \). The value \( \frac{E_0}{2} \) will intersect the graph at two points: one on the left side of the maximum and one on the right side of the maximum.

### Step 5: Count the Intersection Points
From the graph, we can see that there are two points where the electric field equals \( \frac{E_0}{2} \) on either side of the maximum. Thus, there are a total of four points where the electric field is \( \frac{E_0}{2} \):
- Two points on the left side of the maximum (let's call them \( X_1 \) and \( X_2 \))
- Two points on the right side of the maximum (let's call them \( X_3 \) and \( X_4 \))

### Conclusion
Therefore, the total number of points on the axis where the electric field is \( \frac{E_0}{2} \) is **four**.

### Final Answer
The number of points on the axis where the magnitude of the electric field is \( \frac{E_0}{2} \) is **4**.

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