Question

A current carrying straight wire AB is shown in the given diagram. Out of X, Y and Z on which point will the strength of magnetic field be maximum and why?

Hint:

For a straight wire, magnetic field is maximum at points on the perpendicular bisector (same distance from both ends). Field strength decreases as we move toward the ends.

Answer 1

Step 1: Recall the formula for magnetic field due to a straight current-carrying wire.
For a straight wire of finite length, the magnetic field at a point at perpendicular distance \( r \) from the wire is given by: \[ B = \frac{\mu_0 I}{4\pi r} (\sin\theta_1 + \sin\theta_2) \] where \( \theta_1 \) and \( \theta_2 \) are the angles subtended by the wire at the point.
Step 2: For a point near the center of the wire.
When the point is opposite the center of the wire (perpendicular bisector), the angles are equal and maximum for a given distance.
Step 3: Analyze points X, Y, Z.

• Point Y is located directly above the center of wire AB (perpendicular bisector).
• Points X and Z are located above the ends of the wire (near A and B respectively).
• All three points are at the same perpendicular distance from the wire (same vertical height).

Step 4: Compare the angles.

• For point Y (center): The wire subtends maximum angles \( \theta_1 \) and \( \theta_2 \) (almost 90° each if wire is long).
• For points X and Z (near ends): The angles subtended are smaller because the point is not symmetrically placed with respect to the wire.

Step 5: Mathematical comparison.
For a wire of length \( L \), at perpendicular distance \( d \):

• At center: \( \sin\theta_1 + \sin\theta_2 = \frac{2 \times (L/2)}{\sqrt{(L/2)^2 + d^2}} = \frac{L}{\sqrt{(L/2)^2 + d^2}} \)
• At end: \( \sin\theta_1 + \sin\theta_2 = 0 + \frac{L}{\sqrt{L^2 + d^2}} = \frac{L}{\sqrt{L^2 + d^2}} \)

Since \( \sqrt{(L/2)^2 + d^2}<\sqrt{L^2 + d^2} \), the value at center is larger.
Step 6: Conclusion.
The magnetic field strength will be maximum at point Y because:

• It lies on the perpendicular bisector of the wire where the angles subtended by the wire are maximum.
• For the same perpendicular distance, the field is strongest at points opposite the center of the wire.

Step 7: Final answer.
The magnetic field strength will be maximum at point Y because it lies on the perpendicular bisector of the wire where the angles subtended by the wire are maximum.