A cylindrical long wire of radius `R` carries a current `I` uniformly distributed over the cross sectional area of the wire. The magnetic field at a distance `x` from the surface inside the wire is

Options

A. `(mu_0I)/(2pi(R-x)`
B. `(mu_0I)/(2pix)`
C. `(mu_0I)/(2pi(R+x))`
D. none of these

Explanation

To find the magnetic field at a distance \( x \) from the surface inside a cylindrical long wire of radius \( R \) carrying a uniformly distributed current \( I \), we can follow these steps:

### Step 1: Identify the distance from the center of the wire
The distance from the center of the wire to the point where we want to calculate the magnetic field is given by:
\[ r = R - x \]
where \( R \) is the radius of the wire and \( x \) is the distance from the surface of the wire.

### Step 2: Calculate the current density
The current density \( J \) is defined as the current per unit area. Since the current \( I \) is uniformly distributed over the cross-sectional area of the wire, we can express it as:
\[ J = \frac{I}{\pi R^2} \]

### Step 3: Calculate the enclosed current
To find the magnetic field at a distance \( r \) from the center, we need to calculate the current enclosed by a circular cross-section of radius \( r \):
\[ I_{\text{enclosed}} = J \cdot A = J \cdot \pi r^2 \]
Substituting the expression for \( J \):
\[ I_{\text{enclosed}} = \left(\frac{I}{\pi R^2}\right) \cdot \pi r^2 = I \cdot \frac{r^2}{R^2} \]

### Step 4: Substitute \( r \) in terms of \( x \)
Now, substitute \( r = R - x \) into the expression for \( I_{\text{enclosed}} \):
\[ I_{\text{enclosed}} = I \cdot \frac{(R - x)^2}{R^2} \]

### Step 5: Apply Ampère's Law
Using Ampère's Law, the magnetic field \( B \) at a distance \( r \) from the center of the wire is given by:
\[ B = \frac{\mu_0 I_{\text{enclosed}}}{2 \pi r} \]
Substituting \( I_{\text{enclosed}} \):
\[ B = \frac{\mu_0}{2 \pi (R - x)} \cdot I \cdot \frac{(R - x)^2}{R^2} \]

### Step 6: Simplify the expression
Now, simplify the expression for \( B \):
\[ B = \frac{\mu_0 I (R - x)}{2 \pi R^2} \]

### Final Answer
Thus, the magnetic field at a distance \( x \) from the surface inside the wire is:
\[ B = \frac{\mu_0 I (R - x)}{2 \pi R^2} \]

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