Question

Assertion-Reason Type Question Assertion (A): The magnetic field at the center of a circular current carrying loop is directly proportional to the current flowing through it.
Reason (R): According to Biot–Savart’s law, magnetic field produced by a current element is proportional to the current element. Choose the correct answer from the options given below:

• A. Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
• B. Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
• C. Assertion is true but Reason is false.
• D. Assertion is false but Reason is true.

Hint:

Remember: \[ B_{\text{circular loop}}=\frac{\mu_0 I}{2R} \] Magnetic field increases with current and decreases with radius.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Assertion-Reason tasks examine whether separate technical facts are correct on their own and if a logical link connects them. In magnetostatics, Biot-Savart's law calculates the total magnetic field ($B$) set up by moving electric charges. It serves as the foundational framework to derive equations for complex geometries, such as circular wire loops.
Step 2: Key Formula or Approach:
1. According to Biot-Savart's law, the magnetic field differential element $dB$ due to an infinitesimal current element $I \, d\vec{l}$ at a distance vector $\vec{r}$ is: $$ dB = \frac{\mu_0}{4\pi} \frac{I (d\vec{l} \times \hat{r})}{r^2} \implies dB \propto I \, dl $$ 2. Integrating this relationship around a full circular loop of radius $R$ gives the total magnetic field intensity directly at the center: $$ B = \frac{\mu_0 I}{2R} $$
Step 3: Detailed Explanation:
Let's analyze the given statements step-by-step: - Evaluating the Assertion (A): The Derived equation for a circular current-carrying wire loop shows that the central magnetic field is $B = \frac{\mu_0 I}{2R}$. Keeping the loop's geometric radius $R$ constant, it is clear that $B \propto I$. This shows that the central magnetic field scales linearly with the current flowing through the loop. Thus, Assertion (A) is completely true. - Evaluating the Reason (R): Biot-Savart's fundamental equation dictates that the localized magnetic field contribution ($dB$) is proportional to the current element ($I \, dl$). Thus, Reason (R) is also completely true. - Evaluating the Logical Link: To find out why the total magnetic field at the loop center depends linearly on the current ($B \propto I$), we integrate the individual segment fields around the circle. Because every single individual segment contribution scales with the current element ($dB \propto I \, dl$), the accumulated sum must scale directly with current as well. Because the localized proportional relationship in Biot-Savart's law explains why the integrated macro-field scales proportionally, the Reason provides the correct explanation for the Assertion. This matches option (A).
Step 4: Final Answer:
Both Assertion and Reason are true and Reason is the correct explanation of Assertion.