Question

A long wire with a small current element of length 1 cm is placed at the origin and carries a current of 10 A along the x-axis. The magnitude of the magnetic field, due to the element, on the y-axis at a distance 0.5 m from it, would be

• A. 4 x 10\(^{-8}\) T
• B. 5 x 10\(^{-8}\) T
• C. 6 x 10\(^{-8}\) T
• D. 2 x 10\(^{-8}\) T

Hint:

In problems involving Biot-Savart Law, first identify the given parameters: \(I, dl, r\), and the angle \(\theta\). Pay close attention to the geometry to correctly determine \(\theta\). For perpendicular axes (like x and y), \(\theta\) is \(90^\circ\). Also, remember the value of the constant \(\frac{\mu_0}{4\pi} = 10^{-7}\) T·m/A for easy calculation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires the application of the Biot-Savart Law to find the magnetic field produced by a small current-carrying element. The Biot-Savart Law gives the magnetic field at a point in space due to a current element.

Step 2: Key Formula or Approach:
The magnitude of the magnetic field \(dB\) due to a current element \(Id\vec{l}\) at a distance \(r\) is given by the Biot-Savart Law: \[ dB = \frac{\mu_0}{4\pi} \frac{I dl \sin\theta}{r^2} \] where \(\mu_0\) is the permeability of free space (\(4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A}\)), \(I\) is the current, \(dl\) is the length of the element, \(r\) is the distance from the element to the point, and \(\theta\) is the angle between the direction of the current element and the position vector to the point.

Step 3: Detailed Explanation:
Given data:
Current, \(I = 10 \, \text{A}\).
Length of the element, \(dl = 1 \, \text{cm} = 0.01 \, \text{m}\).
Distance to the point on the y-axis, \(r = 0.5 \, \text{m}\).
The current element is along the x-axis, and the point is on the y-axis. Therefore, the angle \(\theta\) between the current element \(d\vec{l}\) and the position vector \(\vec{r}\) is \(90^\circ\). Thus, \(\sin\theta = \sin(90^\circ) = 1\).
The constant \(\frac{\mu_0}{4\pi} = 10^{-7} \, \text{T}\cdot\text{m/A}\).
Calculation:
Substitute the values into the Biot-Savart Law formula: \[ dB = (10^{-7}) \frac{(10 \, \text{A}) \times (0.01 \, \text{m}) \times 1}{(0.5 \, \text{m})^2} \] \[ dB = 10^{-7} \frac{0.1}{0.25} \] \[ dB = 10^{-7} \times 0.4 \] \[ dB = 4 \times 10^{-8} \, \text{T} \]

Step 4: Final Answer:
The magnitude of the magnetic field due to the element at the given point is \(4 \times 10^{-8}\) T.