A straight long wire carrying current 50 A is placed on a horizontal surface. Another wire of unit length carrying 22 A current is balanced in air above at a distance of` 0.5` cm from the long wire. Calculate the mass of the wire.

Answer

To solve the problem, we need to calculate the mass of the wire carrying a current of 22 A that is balanced in air above a long wire carrying a current of 50 A. Here's a step-by-step solution:

### Step 1: Understand the Forces Acting on the Wire
The wire carrying 22 A is balanced in the air due to the magnetic force exerted by the long wire carrying 50 A. The downward force acting on the wire is its weight, which can be expressed as:
\[ F_{\text{down}} = mg \]
where \( m \) is the mass of the wire and \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)).

### Step 2: Calculate the Magnetic Field (B) Due to the Long Wire
The magnetic field \( B \) at a distance \( d \) from a long straight wire carrying a current \( I_1 \) is given by the formula:
\[ B = \frac{\mu_0 I_1}{2\pi d} \]
where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \) (the permeability of free space), \( I_1 = 50 \, \text{A} \), and \( d = 0.5 \, \text{cm} = 0.005 \, \text{m} \).

### Step 3: Substitute Values to Calculate B
Substituting the values into the formula:
\[ B = \frac{4\pi \times 10^{-7} \times 50}{2\pi \times 0.005} \]
Simplifying this gives:
\[ B = \frac{4 \times 10^{-7} \times 50}{0.01} = 2 \times 10^{-3} \, \text{T} \]

### Step 4: Calculate the Magnetic Force (F) on the Wire
The magnetic force \( F \) acting on the wire carrying current \( I_2 \) (22 A) of length \( l \) (1 m) in the magnetic field \( B \) is given by:
\[ F = I_2 l B \]
Substituting the values:
\[ F = 22 \times 1 \times 2 \times 10^{-3} = 44 \times 10^{-3} \, \text{N} \]

### Step 5: Set the Forces Equal for Balance
For the wire to be balanced, the upward magnetic force must equal the downward gravitational force:
\[ mg = F \]
Substituting for \( F \):
\[ mg = 44 \times 10^{-3} \]

### Step 6: Solve for Mass (m)
Rearranging the equation to solve for \( m \):
\[ m = \frac{F}{g} = \frac{44 \times 10^{-3}}{10} = 4.4 \times 10^{-3} \, \text{kg} \]

### Step 7: Convert to Grams
To convert the mass into grams:
\[ m = 4.4 \, \text{grams} \]

### Final Answer
The mass of the wire is **4.4 grams**.

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