Question

A long straight wire of a circular cross-section with radius \( a \) carries a steady current \( I \). The current is uniformly distributed across this cross-section. The plot of magnitude of magnetic field \( B \) with distance \( r \) from the centre of the wire is given by:

• A. \[ \text{Plot 1: } \] 
  
• B. \[ \text{Plot 2: } \] 
  
• C. \[ \text{Plot 3: } \]
  
• D. \[ \text{Plot 4: } \] 
  

Hint:

The magnetic field inside a current-carrying wire increases linearly with radius, while outside, the field decreases inversely with distance. Use this information to select the correct plot.

Answer 1

The Correct Option is C

In the case of a long straight wire with a uniformly distributed current, the magnetic field outside the wire decreases inversely with the distance from the wire (\( B \propto \frac{1}{r} \)) as given by Ampere's law. Inside the wire, the magnetic field increases linearly with the distance from the centre (\( B \propto r \)).

The correct plot represents a magnetic field that increases linearly within the radius of the wire and then decreases inversely beyond the radius.

Final Answer: Plot 3.

Answer 2

Step 1: Understand the setup.
We are dealing with a long straight wire of circular cross-section with radius \( a \), which carries a steady current \( I \). The current is uniformly distributed across this cross-section. We need to determine the relationship between the magnitude of the magnetic field \( B \) and the radial distance \( r \) from the center of the wire.

Step 2: Magnetic field inside the wire (for \( r < a \)).
For a current-carrying wire with a uniform current distribution, the magnetic field inside the wire (for \( r < a \)) increases linearly with the radial distance \( r \). This can be expressed as:
\[ B(r) = \frac{\mu_0 I r}{2\pi a^2}. \]
Here, \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( a \) is the radius of the wire.

Step 3: Magnetic field outside the wire (for \( r > a \)).
Outside the wire (for \( r > a \)), the magnetic field decreases with the inverse of the distance \( r \). It follows the relationship:
\[ B(r) = \frac{\mu_0 I}{2\pi r}. \]
As \( r \) increases, the magnetic field decreases in proportion to \( \frac{1}{r} \).

Step 4: Analysis of the plot.
The plot provided shows the magnitude of the magnetic field \( B \) as a function of the radial distance \( r \). Key features:
- For \( r < a \), the magnetic field increases linearly with \( r \).
- For \( r > a \), the magnetic field decreases with \( \frac{1}{r} \).
- The value of \( B \) at the surface of the wire (when \( r = a \)) is \( B_0 \), and as \( r \) increases, the field decreases following the \( \frac{1}{r} \) dependence.

Final Answer:
The plot shown represents the correct relationship between the magnetic field \( B \) and the distance \( r \) for a long straight wire carrying a steady current.