Question

What is the value of the line integral of a magnetic field around a closed loop according to Ampère’s Circuital Law?

• A. \(0\)
• B. \(\mu_0 I\)
• C. \(\dfrac{I}{\mu_0}\)
• D. \(B \times I\)

Hint:

{Ampère’s Law:} \(\displaystyle \oint \vec{B}\cdot d\vec{l} = \mu_0 I\). The circulation of magnetic field around a closed loop depends on the enclosed current.

Answer 1

The Correct Option is B

Ampère's Circuital Law is a fundamental principle in electromagnetism which relates the integrated magnetic field around a closed loop to the electric current passing through the loop. The law is mathematically expressed as:

\(\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}\)

Here,

• \(\mathbf{B}\) is the magnetic field.
• \(d\mathbf{l}\) is a differential element of the loop path.
• \(\mu_0\) is the permeability of free space, a constant.
• \(I_{enc}\) is the total current enclosed by the loop.

The statement of Ampère’s Circuital Law tells us that the line integral of the magnetic field around a closed path is directly proportional to the total current enclosed by the path. Therefore, according to Ampère's Circuital Law, the value of the line integral of a magnetic field around a closed loop is:

Correct Answer: \(\mu_0 I\)

Let's analyze why this is the correct option and rule out the rest:

• \(0\): This would imply no enclosed current, which contradicts the assumption that there is a current flowing through the loop.
• \(\dfrac{I}{\mu_0}\): This is contrary to the formulation of Ampère's Law, which involves multiplication by \(\mu_0\), not division.
• \(B \times I\): This option suggests a product irrelevant to the integral definition given by Ampère’s Law.

In conclusion, the correct value for the line integral of the magnetic field around a closed loop, according to Ampère's Circuital Law, is \(\mu_0 I\).