A current of 10 A is flowing through a circular coil of 5 turns, each of radius 7 cm. The coil lies in the X-Y plane. What is the magnitude and direction of the magnetic dipole moment associated with it ?

Options

A. `0.47 Am^(2)` along Z axis
B. `0.77 Am^(2)` along Z axis
C. `0.77 Am^(2)` along Y axis
D. `1.54 Am^(2)` along X axis

Answer

To find the magnitude and direction of the magnetic dipole moment associated with a circular coil, we can follow these steps:

### Step 1: Understand the Formula for Magnetic Dipole Moment
The magnetic dipole moment (μ) for a coil is given by the formula:
\[ \mu = n \cdot I \cdot A \]
where:
- \( n \) = number of turns in the coil
- \( I \) = current flowing through the coil (in Amperes)
- \( A \) = area of the coil (in square meters)

### Step 2: Calculate the Area of the Coil
The area \( A \) of a circular coil is calculated using the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the coil. Given that the radius is 7 cm, we need to convert it to meters:
\[ r = 7 \, \text{cm} = \frac{7}{100} \, \text{m} = 0.07 \, \text{m} \]
Now, substituting the value of \( r \) into the area formula:
\[ A = \pi (0.07)^2 = \pi \times 0.0049 \, \text{m}^2 \approx 0.0154 \, \text{m}^2 \]

### Step 3: Substitute Values into the Magnetic Dipole Moment Formula
Now, we can substitute the values into the magnetic dipole moment formula:
- \( n = 5 \) (number of turns)
- \( I = 10 \, \text{A} \) (current)
- \( A \approx 0.0154 \, \text{m}^2 \)

Calculating \( \mu \):
\[ \mu = n \cdot I \cdot A = 5 \cdot 10 \cdot 0.0154 \approx 0.77 \, \text{A m}^2 \]

### Step 4: Determine the Direction of the Magnetic Dipole Moment
The direction of the magnetic dipole moment is determined by the right-hand rule. When the fingers of the right hand curl in the direction of the current flow, the thumb points in the direction of the magnetic dipole moment. Since the coil lies in the X-Y plane, the magnetic dipole moment will point along the Z-axis.

### Final Answer
The magnitude of the magnetic dipole moment is approximately \( 0.77 \, \text{A m}^2 \) and its direction is along the Z-axis.

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