Find the current density in a wire of diameter `2 xx 10^(-10)`m in which current of 5 A flows ?

Options

A. `1.7 xx 10^(-10)`
B. `3.187 xx 10^(-27)`
C. `0.4 xx 10^(6)`
D. `16.7 xx 10^(-10)`

Explanation

To find the current density in a wire, we can use the formula:

\[ J = \frac{I}{A} \]

where \( J \) is the current density, \( I \) is the current flowing through the wire, and \( A \) is the cross-sectional area of the wire.

### Step 1: Calculate the cross-sectional area of the wire

The wire has a circular cross-section, and the area \( A \) can be calculated using the formula for the area of a circle:

\[ A = \frac{\pi d^2}{4} \]

where \( d \) is the diameter of the wire.

Given that the diameter \( d = 2 \times 10^{-10} \) m, we can substitute this value into the formula:

\[ A = \frac{\pi (2 \times 10^{-10})^2}{4} \]

### Step 2: Calculate the area

Calculating \( (2 \times 10^{-10})^2 \):

\[ (2 \times 10^{-10})^2 = 4 \times 10^{-20} \]

Now substituting this back into the area formula:

\[ A = \frac{\pi \times 4 \times 10^{-20}}{4} \]

The \( 4 \) in the numerator and denominator cancels out:

\[ A = \pi \times 10^{-20} \text{ m}^2 \]

### Step 3: Substitute the values into the current density formula

Now we know the current \( I = 5 \) A. We can substitute \( I \) and \( A \) into the current density formula:

\[ J = \frac{I}{A} = \frac{5}{\pi \times 10^{-20}} \]

### Step 4: Calculate the current density

Using \( \pi \approx 3.14 \):

\[ J = \frac{5}{3.14 \times 10^{-20}} \approx \frac{5}{3.14} \times 10^{20} \]

Calculating \( \frac{5}{3.14} \):

\[ \frac{5}{3.14} \approx 1.59 \]

Thus,

\[ J \approx 1.59 \times 10^{20} \text{ A/m}^2 \]

### Final Answer

The current density in the wire is approximately:

\[ J \approx 1.59 \times 10^{20} \text{ A/m}^2 \]

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