Question

The pattern of magnetic field due to a current carrying wire depends upon the shape made by that wire. Justify.

Hint:

Biot-Savart law: Field depends on distance, angle, and current element direction. Different shapes \(\Rightarrow\) different distributions of elements \(\Rightarrow\) different field patterns.

Answer 1

Step 1: Recall the Biot-Savart law.
According to Biot-Savart law, the magnetic field \( dB \) due to a small current element \( Id\vec{l} \) at a point is given by: \[ dB = \frac{\mu_0}{4\pi} \frac{I \, dl \sin\theta}{r^2} \] where \( r \) is the distance from the element to the point, and \( \theta \) is the angle between the current element and the position vector.
Step 2: Dependence on shape.
The total magnetic field at a point is obtained by integrating contributions from all current elements along the wire. The shape of the wire determines:

• The direction of each current element \( d\vec{l} \)
• The distance \( r \) from each element to the point
• The angle \( \theta \) for each element

Step 3: Examples of different shapes.

• Straight wire: Magnetic field lines are concentric circles around the wire. Field strength decreases with distance.
• Circular loop: Field lines are concentrated at the center, producing a uniform field at the center. Field pattern resembles that of a bar magnet.
• Solenoid: Produces a nearly uniform magnetic field inside and field pattern similar to a bar magnet outside.
• Helical wire: Complex field pattern with both axial and circular components.

Step 4: Final justification.
The magnetic field pattern depends on the wire's shape because the direction and magnitude of the field at any point is the vector sum of contributions from all current elements, whose positions and orientations are determined by the wire's geometry.