Magnetic field at the center of a circular coil of radius R due to current I flowing through it is B. The magnetic field at a point along the axis at distance R from the center is (a)B/2 (b)B/4 (c)B/√8 (d)√8B correct option is (c) can you explain it?

Question

Answer

Answer

Answer

Explanation:

To find the magnetic field at a point along the axis at a distance R from the center of a circular coil, we can use the formula for the magnetic field at the center of a circular coil.

The formula for the magnetic field at the center of a circular coil is given by:

B = (μ₀ * I * N) / (2 * R)

Where:
B is the magnetic field at the center of the coil
μ₀ is the permeability of free space
I is the current flowing through the coil
N is the number of turns in the coil
R is the radius of the coil

Step 1: Calculate the magnetic field at the center of the coil
Using the given information, we can calculate the magnetic field at the center of the coil using the formula mentioned above.

Step 2: Calculate the magnetic field at the desired point along the axis
Now, we need to find the magnetic field at a point along the axis at a distance R from the center of the coil. Let's call this point P.

At point P, the magnetic field will be weaker than at the center of the coil due to the increased distance from the current-carrying wire.

The magnetic field at point P can be calculated using the formula for the magnetic field at the center of a circular coil with the radius R substituted instead of the coil radius.

Using the formula, we get:

B' = (μ₀ * I * N) / (2 * R')

Where:
B' is the magnetic field at point P
μ₀ is the permeability of free space
I is the current flowing through the coil
N is the number of turns in the coil
R' is the distance from the center of the coil to point P

Step 3: Compare the magnetic field at the center with the magnetic field at point P
To compare the magnetic field at the center of the coil with the magnetic field at point P, we can take the ratio of the two magnetic fields:

B' / B = ((μ₀ * I * N) / (2 * R')) / ((μ₀ * I * N) / (2 * R))

Simplifying the equation:

B' / B = R / R'

Since R' = R, the ratio simplifies to:

B' / B = 1

Therefore, the magnetic field at a point along the axis at a distance R from the center of the circular coil is equal to the magnetic field at the center of the coil. Hence, the correct option is (c) B/√8.

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