Step 1: Understanding the Assertion (A):
The assertion states that the magnetic field at the center of a current-carrying circular coil is directly proportional to the number of turns (N) in the coil. This is given by the formula:
\[
B = \frac{\mu_0 NI}{2r}
\]
where:
- \( B \) is the magnetic field at the center of the coil,
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current flowing through the coil,
- \( r \) is the radius of the coil, and
- \( N \) is the number of turns in the coil.
According to this formula, as \( N \) increases, the magnetic field \( B \) increases as well, because the magnetic field is directly proportional to the number of turns.
Step 2: Understanding the Reason (R):
The reason provides an explanation for why the magnetic field is proportional to the number of turns. It states that since the current in each turn is the same and the magnetic field due to each turn is in the same direction, the total magnetic field is the sum of the individual magnetic fields generated by each turn. As a result, the more turns there are, the greater the cumulative magnetic field at the center of the coil.
Step 3: Connecting Assertion (A) and Reason (R):
- The assertion tells us that the magnetic field at the center of the coil is proportional to the number of turns. This is mathematically expressed in the given formula.
- The reason explains how this happens: since the current in each turn is the same and the magnetic fields from each turn add together, the total magnetic field increases with the number of turns. This directly supports the assertion.
Step 4: Conclusion:
Both the assertion and the reason are true, and the reason correctly explains the assertion. The increase in the number of turns results in an increased magnetic field at the center of the coil, as described by the formula \( B = \frac{\mu_0 NI}{2r} \).