Question

A long straight wire of circular cross-section (radius \( a \)) is carrying a steady current \( I \). The current \( I \) is uniformly distributed across this cross-section. The magnetic field is:

• A. Zero in the region \( r<a \) and inversely proportional to \( r \) in the region \( r>a \)
• B. Inversely proportional to \( r \) in the region \( r<a \) and uniform throughout in the region \( r>a \)
• C. Directly proportional to \( r \) in the region \( r<a \) and inversely proportional to \( r \) in the region \( r>a \)
• D. Uniform in the region \( r<a \) and inversely proportional to distance \( r \) from the axis, in the region \( r>a \)

Hint:

For a long straight wire with uniformly distributed current:
Inside the wire (\( r<a \)), \( B \propto r \).
Outside the wire (\( r>a \)), \( B \propto \frac{1}{r} \).

Answer 1

The Correct Option is C

Step 1: Understanding the Magnetic Field Distribution
The magnetic field \( B \) around a current
carrying wire is given by Ampère's Circuital Law: \[ \oint B \cdot dl = \mu_0 I_{\text{enc}} \] where \( I_{\text{enc}} \) is the current enclosed by the chosen circular Amperian loop.
Step 2: Magnetic Field Inside the Wire (\( r<a \))
Since the current is uniformly distributed, the enclosed current at a radius \( r \) (where \( r<a \)) is: \[ I_{\text{enc}} = I \frac{\pi r^2}{\pi a^2} = I \frac{r^2}{a^2} \] Applying Ampère’s law for a circular path of radius \( r \): \[ B \cdot 2\pi r = \mu_0 I_{\text{enc}} \] Substituting \( I_{\text{enc}} \): \[ B \cdot 2\pi r = \mu_0 I \frac{r^2}{a^2} \] \[ B = \frac{\mu_0 I r}{2\pi a^2} \] This shows that the magnetic field inside the wire is directly proportional to \( r \).
Step 3: Magnetic Field Outside the Wire (\( r>a \))
For \( r>a \), the entire current \( I \) is enclosed within the Amperian loop: \[ B \cdot 2\pi r = \mu_0 I \] \[ B = \frac{\mu_0 I}{2\pi r} \] Thus, the magnetic field outside the wire is inversely proportional to \( r \). Final Answer:
The magnetic field inside the wire is directly proportional to \( r \), and outside the wire, it is inversely proportional to \( r \).