A metallic wire having length of 2 m and weight of `4xx10^(-3)` N is found to remain at rest in a uniform and transverse magnetic field of `2xx10^(-4) T`. Current flowing through the wire is :

Options

A. 10A
B. 5A
C. 2A
D. 1A

Answer

To solve the problem, we need to find the current flowing through a metallic wire that remains at rest in a uniform and transverse magnetic field. Here's the step-by-step solution:

### Step 1: Understand the Forces Acting on the Wire
Since the wire is at rest, the net force acting on it must be zero. This means that the weight of the wire (W) is balanced by the magnetic force (F) acting on it.

### Step 2: Write the Equation for Forces
The weight of the wire (W) can be expressed as:
\[ W = m \cdot g \]
where \( m \) is the mass of the wire and \( g \) is the acceleration due to gravity. However, we are given the weight directly:
\[ W = 4 \times 10^{-3} \, \text{N} \]

The magnetic force (F) on the wire in a magnetic field can be expressed as:
\[ F = B \cdot I \cdot L \]
where:
- \( B \) is the magnetic field strength,
- \( I \) is the current flowing through the wire,
- \( L \) is the length of the wire.

### Step 3: Set the Forces Equal
Since the wire is at rest, we can set the weight equal to the magnetic force:
\[ W = F \]
\[ 4 \times 10^{-3} \, \text{N} = B \cdot I \cdot L \]

### Step 4: Substitute Known Values
We know:
- \( B = 2 \times 10^{-4} \, \text{T} \)
- \( L = 2 \, \text{m} \)

Substituting these values into the equation:
\[ 4 \times 10^{-3} = (2 \times 10^{-4}) \cdot I \cdot 2 \]

### Step 5: Simplify the Equation
Now, simplify the equation:
\[ 4 \times 10^{-3} = 4 \times 10^{-4} \cdot I \]

### Step 6: Solve for Current (I)
To find \( I \), divide both sides by \( 4 \times 10^{-4} \):
\[ I = \frac{4 \times 10^{-3}}{4 \times 10^{-4}} \]
\[ I = 10 \, \text{A} \]

### Final Answer
The current flowing through the wire is:
\[ I = 10 \, \text{A} \]