Curves in the graph shown in Fig. give, as function of radius distance r, the magnitude B of the magnetic field inside and outside four long wire a,b,c and d, carrying currents that are uniformly distributed across the cross sections of the wires. Overlapping portions of the plots are indicated by double labels.

The current density in wire a is

A. greater than in wire `c`.
B. less than in wire `c`.
C. equal to that in wire `c`.
D. not comparable to that of in wire `c` due to lack of information

Options

A. greater than in wire `c`.
B. less than in wire `c`.
C. equal to that in wire `c`.
D. not comparable to that of in wire `c` due to lack of information

Answer

To solve the question regarding the current density in wire A compared to wire C, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Magnetic Field Inside a Wire**:
The magnetic field \( B \) inside a long straight wire carrying a uniformly distributed current can be expressed as:
\[ B = \frac{\mu_0 I r}{2 \pi R^2} \]
where \( I \) is the current, \( r \) is the radial distance from the center of the wire, \( R \) is the radius of the wire, and \( \mu_0 \) is the permeability of free space.

2. **Relating Magnetic Field to Current Density**:
The current density \( J \) in a wire is defined as:
\[ J = \frac{I}{A} \]
where \( A \) is the cross-sectional area of the wire. For a circular wire, \( A = \pi R^2 \), thus:
\[ J = \frac{I}{\pi R^2} \]

3. **Finding the Slope of the Magnetic Field Graph**:
The slope of the graph of \( B \) versus \( r \) (the radial distance) inside the wire is proportional to the current density. The slope can be represented as:
\[ \text{slope} = \frac{dB}{dr} \]
This slope is directly related to the current density \( J \).

4. **Comparing the Slopes for Wires A and C**:
From the graph, we can observe the slopes corresponding to wires A and C. If the slope of wire A is greater than that of wire C, it indicates that the current density in wire A is greater than in wire C.

5. **Conclusion**:
Since the slope of wire A is greater than the slope of wire C, we conclude that the current density in wire A is greater than that in wire C. Therefore, the correct answer is:
\[ \text{A. greater than in wire C.} \]