A metal ring of radius `r =0.5m` with its plane normal a unifrom magnetic field `B` of induction `0.2T` carries a current `I=100A` The tension in newtons developed in the ring is

Options

A. (a)100
B. (b)50
C. (c)25
D. (d)10

Explanation

To solve the problem of finding the tension developed in the metal ring, we will follow these steps:

### Step 1: Understand the setup
We have a metal ring of radius \( r = 0.5 \, \text{m} \) placed in a uniform magnetic field \( B = 0.2 \, \text{T} \). The ring carries a current \( I = 100 \, \text{A} \).

### Step 2: Calculate the magnetic force on the ring
The magnetic force \( F \) on a current-carrying conductor in a magnetic field can be calculated using the formula:
\[ F = I \cdot L \cdot B \cdot \sin(\theta) \]
where:
- \( I \) is the current,
- \( L \) is the length of the conductor,
- \( B \) is the magnetic field strength,
- \( \theta \) is the angle between the direction of the current and the magnetic field.

In this case, the current flows around the ring, and the magnetic field is perpendicular to the plane of the ring, so \( \theta = 90^\circ \) and \( \sin(90^\circ) = 1 \).

### Step 3: Calculate the length of the ring
The length \( L \) of the ring (which is the circumference of the circle) is given by:
\[ L = 2\pi r = 2\pi \times 0.5 = \pi \, \text{m} \]

### Step 4: Substitute values into the force equation
Now substituting the values into the force equation:
\[ F = I \cdot L \cdot B = 100 \, \text{A} \cdot \pi \, \text{m} \cdot 0.2 \, \text{T} \]
\[ F = 100 \cdot \pi \cdot 0.2 = 20\pi \, \text{N} \]

### Step 5: Calculate the tension in the ring
The tension \( T \) in the ring must balance the total magnetic force acting on the ring. Since the force acts uniformly around the ring, we can relate the total magnetic force to the tension:
\[ 2T = F \]
Thus,
\[ T = \frac{F}{2} = \frac{20\pi}{2} = 10\pi \, \text{N} \]

### Step 6: Final answer
Using the approximate value of \( \pi \approx 3.14 \):
\[ T \approx 10 \times 3.14 = 31.4 \, \text{N} \]

### Conclusion
The tension developed in the ring is approximately \( 31.4 \, \text{N} \).

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