A magnetic field of `5xx10^(-5)` T is produced at a perpendicular distance of `0.2` m from a long straight wire carrying electric current. If the permeability of free space is `4 pi xx 10^(-7)T m//A,` the current passing through the wire in A is

Options

A. 0.2 A
B. 0 A
C. 4 A
D. 50 A

Answer

To find the current passing through the wire, we will use the formula for the magnetic field around a long straight wire, which is given by:

\[ B = \frac{\mu_0 I}{2 \pi r} \]

Where:
- \( B \) is the magnetic field (in Tesla),
- \( \mu_0 \) is the permeability of free space (\( 4 \pi \times 10^{-7} \, \text{T m/A} \)),
- \( I \) is the current (in Amperes),
- \( r \) is the perpendicular distance from the wire (in meters).

### Step 1: Rearranging the formula to solve for current \( I \)

We can rearrange the formula to solve for the current \( I \):

\[ I = \frac{B \cdot 2 \pi r}{\mu_0} \]

### Step 2: Substitute the known values

Now, we will substitute the known values into the equation:
- \( B = 5 \times 10^{-5} \, \text{T} \)
- \( r = 0.2 \, \text{m} \)
- \( \mu_0 = 4 \pi \times 10^{-7} \, \text{T m/A} \)

Substituting these values into the equation gives:

\[ I = \frac{(5 \times 10^{-5}) \cdot (2 \pi \cdot 0.2)}{4 \pi \times 10^{-7}} \]

### Step 3: Simplifying the equation

We can simplify the equation:

1. The \( \pi \) in the numerator and denominator cancels out:

\[ I = \frac{(5 \times 10^{-5}) \cdot (2 \cdot 0.2)}{4 \times 10^{-7}} \]

2. Calculate \( 2 \cdot 0.2 = 0.4 \):

\[ I = \frac{(5 \times 10^{-5}) \cdot 0.4}{4 \times 10^{-7}} \]

3. Calculate \( 5 \times 0.4 = 2 \):

\[ I = \frac{2 \times 10^{-5}}{4 \times 10^{-7}} \]

### Step 4: Further simplification

Now, we can simplify \( \frac{2 \times 10^{-5}}{4 \times 10^{-7}} \):

1. Divide \( 2 \) by \( 4 \):

\[ I = \frac{1}{2} \times 10^{2} = 0.5 \times 10^{2} \]

2. This simplifies to:

\[ I = 50 \, \text{A} \]

### Final Answer

Thus, the current passing through the wire is:

\[ I = 50 \, \text{A} \]