The magnetic field at the centre of a circular coil of radius r, due to current I flowing through it, is B. The magnetic field at a point along the axis at a distance `r/2` from the centre

Options

A. B/2
B. 2B
C. `(2/sqrt5)^3B`
D. `(2/sqrt3)^3B`

Answer

To solve the problem of finding the magnetic field at a point along the axis of a circular coil at a distance \( \frac{r}{2} \) from its center, we can follow these steps:

### Step 1: Understand the Magnetic Field at the Center of the Coil
The magnetic field \( B \) at the center of a circular coil of radius \( r \) carrying a current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2r} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Identify the Point of Interest
We need to find the magnetic field at a point \( P \) along the axis of the coil, at a distance \( \frac{r}{2} \) from the center of the coil.

### Step 3: Use the Formula for Magnetic Field on the Axis of a Circular Coil
The magnetic field \( B_P \) at a distance \( x \) from the center of a circular coil along its axis is given by:
\[ B_P = \frac{\mu_0 I r^2}{2(r^2 + x^2)^{3/2}} \]
In our case, \( x = \frac{r}{2} \).

### Step 4: Substitute Values into the Formula
Now, substituting \( x = \frac{r}{2} \) into the formula:
\[ B_P = \frac{\mu_0 I r^2}{2\left(r^2 + \left(\frac{r}{2}\right)^2\right)^{3/2}} \]
Calculating \( r^2 + \left(\frac{r}{2}\right)^2 \):
\[ r^2 + \left(\frac{r}{2}\right)^2 = r^2 + \frac{r^2}{4} = \frac{4r^2 + r^2}{4} = \frac{5r^2}{4} \]

### Step 5: Calculate \( (r^2 + x^2)^{3/2} \)
Now, we need to calculate \( \left(\frac{5r^2}{4}\right)^{3/2} \):
\[ \left(\frac{5r^2}{4}\right)^{3/2} = \left(\frac{5^{3/2} r^3}{8}\right) = \frac{5\sqrt{5} r^3}{8} \]

### Step 6: Substitute Back into the Magnetic Field Formula
Now substitute this back into the expression for \( B_P \):
\[ B_P = \frac{\mu_0 I r^2}{2 \cdot \frac{5\sqrt{5} r^3}{8}} = \frac{\mu_0 I r^2 \cdot 8}{10\sqrt{5} r^3} \]
This simplifies to:
\[ B_P = \frac{4\mu_0 I}{5\sqrt{5} r} \]

### Step 7: Relate \( B_P \) to \( B \)
Since we know that \( B = \frac{\mu_0 I}{2r} \), we can express \( B_P \) in terms of \( B \):
\[ B_P = \frac{4}{5\sqrt{5}} B \]

### Final Answer
Thus, the magnetic field at the point along the axis at a distance \( \frac{r}{2} \) from the center of the coil is:
\[ B_P = \frac{4}{5\sqrt{5}} B \]