The ratio of magnetic fileds of a dipole at a distance d in the broadside - on position to end - on position is .

Options

A. Zero
B. infintiy
C. 2
D. `(1)/(2)`

Explanation

To find the ratio of the magnetic fields of a dipole at a distance \( d \) in the broadside-on position to the end-on position, we will follow these steps:

### Step 1: Understand the Magnetic Field of a Dipole
The magnetic field \( B \) at a point due to a magnetic dipole can be expressed using the formula:
\[ B = \frac{\mu_0}{4\pi} \frac{2M \cos \theta}{r^3} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( M \) is the magnetic moment of the dipole,
- \( r \) is the distance from the dipole to the point where the field is being measured,
- \( \theta \) is the angle between the dipole moment and the line connecting the dipole to the point.

### Step 2: Calculate the Magnetic Field in the Broadside-On Position
In the broadside-on position, the angle \( \theta \) is \( 90^\circ \):
\[ B_{\text{broadside}} = \frac{\mu_0}{4\pi} \frac{M}{d^3} \cdot (1 + 3 \cos^2 90^\circ) \]
Since \( \cos 90^\circ = 0 \):
\[ B_{\text{broadside}} = \frac{\mu_0}{4\pi} \frac{M}{d^3} \cdot 1 = \frac{\mu_0 M}{4\pi d^3} \]

### Step 3: Calculate the Magnetic Field in the End-On Position
In the end-on position, the angle \( \theta \) is \( 0^\circ \):
\[ B_{\text{end}} = \frac{\mu_0}{4\pi} \frac{M}{d^3} \cdot (1 + 3 \cos^2 0^\circ) \]
Since \( \cos 0^\circ = 1 \):
\[ B_{\text{end}} = \frac{\mu_0}{4\pi} \frac{M}{d^3} \cdot (1 + 3 \cdot 1) = \frac{\mu_0}{4\pi} \frac{M}{d^3} \cdot 4 = \frac{4\mu_0 M}{4\pi d^3} = \frac{\mu_0 M}{\pi d^3} \]

### Step 4: Calculate the Ratio of the Magnetic Fields
Now, we can find the ratio of the magnetic fields in the broadside-on position to the end-on position:
\[ \text{Ratio} = \frac{B_{\text{broadside}}}{B_{\text{end}}} = \frac{\frac{\mu_0 M}{4\pi d^3}}{\frac{\mu_0 M}{\pi d^3}} = \frac{1}{4} \]

### Final Answer
Thus, the ratio of the magnetic fields of a dipole at a distance \( d \) in the broadside-on position to the end-on position is:
\[ \frac{1}{4} \]

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