Question

The straight wire AB carries a current \(I\). The ends of the wire subtend angles \(\theta_1\) and \(\theta_2\) at the point \(P\) as shown in the figure. The magnetic field at the point \(P\) is: 
 

 

• A. \(\frac{\mu_0 I}{4\pi d} (\sin \theta_1 - \sin \theta_2)\)
• B. \(\frac{\mu_0 I}{4\pi d} (\sin \theta_1 + \sin \theta_2)\)
• C. \(\frac{\mu_0 I}{4\pi d} (\cos \theta_1 - \cos \theta_2)\)
• D. \(\frac{\mu_0 I}{4\pi d} (\cos \theta_1 + \cos \theta_2)\)

Hint:

The Biot-Savart law is fundamental for calculating the magnetic field due to current-carrying elements. For a straight wire, the field depends on the angles formed between the wire and the point where the field is being calculated.

Answer 1

The Correct Option is A

This problem requires calculating the magnetic field at point \( P \) from a straight wire carrying current. The wire's endpoints form angles \( \alpha \) and \( \beta \) with point \( P \). The Biot-Savart law is employed to derive the solution, which describes the magnetic field generated by a current element.
Step 1: {Understanding the Biot-Savart Law} 
The Biot-Savart law defines the magnetic field \( d\mathbf{B} \) at a point due to a minute current element \( I \, d\mathbf{l} \) as follows: \[ d\mathbf{B} = \frac{\mu_0 I}{4 \pi} \frac{d\mathbf{l} \times \hat{r}}{r^2} \] Key variables include: - \( \mu_0 \): permeability of free space, - \( I \): current, - \( d\mathbf{l} \): infinitesimal length of the wire element, - \( \hat{r} \): unit vector from the wire element to the calculation point, - \( r \): distance from the wire element to the calculation point. 
Step 2: {Applying the Biot-Savart Law to a Straight Wire} 
To determine the magnetic field at point \( P \) for a straight current-carrying wire, integration of contributions from all infinitesimal wire elements is necessary. The resultant magnetic field for a finite straight wire at point \( P \) is expressed as: \[ B = \frac{\mu_0 I}{4 \pi d} (\sin \theta_1 - \sin \theta_2) \] Here: - \( d \) is the perpendicular distance from the wire to point \( P \), - \( \theta_1 \) and \( \theta_2 \) are the angles between the line connecting point \( P \) to the wire ends and the wire itself. This formula is derived through integration of the Biot-Savart law along the wire. The terms \( \sin \theta_1 \) and \( \sin \theta_2 \) arise from the geometric configuration of the current element contributions to the magnetic field. 
Step 3: {Conclusion} 
Consequently, the magnetic field at point \( P \) due to a straight current-carrying wire, where the wire ends form angles \( \alpha \) and \( \beta \) with the point, is: \[ B = \frac{\mu_0 I}{4 \pi d} (\sin \theta_1 - \sin \theta_2) \] Option (A) is therefore the correct choice.