1)The field normal to the plane of a wire of n turns and radius r which carries a current I is measured on the axis of the coil at a small distance h from the centre of the coil.This is smaller than the field at the centre by the fraction.

a) (3h^2)/(2r^2) b) (2h^2)/(3r^2) c) (3r^2)/(2h^2) d) (2r^2)/(3h^2)

2)The magnetic field inside a current carrying long wire of circular cross section varies as(r:distance from center)

a) r^0 b)r c)r^2 d)r^3

Let's break down your questions about magnetic fields in a systematic way. Both of these questions relate to the behavior of magnetic fields generated by current-carrying conductors, which is a fundamental concept in electromagnetism.

Magnetic Field of a Coil

For the first question, we need to analyze the magnetic field produced by a circular coil of wire with n turns and radius r, carrying a current I. When we measure the magnetic field along the axis of the coil at a small distance h from its center, we can use the formula for the magnetic field along the axis of a circular coil.

The magnetic field B at a distance h from the center of the coil can be approximated using the formula:

• B(h) = (μ₀ * n * I) / (2 * r²) * (1 - (3h²)/(2r²))

At the center of the coil (h = 0), the magnetic field is:

• B(0) = (μ₀ * n * I) / (2 * r)

To find how the field at distance h compares to the field at the center, we look at the fraction by which the field at h is smaller than at the center:

• Fraction = (B(0) - B(h)) / B(0)

After substituting the expressions for B(h) and B(0), we find that the field at distance h is smaller than the field at the center by the fraction:

• (3h²)/(2r²)

Thus, the correct answer is (3h²)/(2r²).

Magnetic Field Inside a Long Wire

Now, let’s discuss the second question regarding the magnetic field inside a long wire with a circular cross-section. The magnetic field B inside a long, straight wire carrying a uniform current varies with the distance r from the center of the wire.

According to Ampère's Law, the magnetic field inside a long wire is directly proportional to the distance from the center of the wire. This can be expressed mathematically as:

• B(r) = (μ₀ * I) / (2π * R²) * r

Here, R is the radius of the wire, and I is the current flowing through it. This indicates that as you move away from the center of the wire (increasing r), the magnetic field strength increases linearly.

Therefore, the magnetic field inside a current-carrying long wire varies as:

• r

So, the correct answer to this question is r.

Summary of Answers

• For the first question, the fraction by which the field at distance h is smaller than at the center is (3h²)/(2r²).
• For the second question, the magnetic field inside a long wire varies as r.

Understanding these principles is crucial for grasping the behavior of magnetic fields in various configurations, which is foundational in both physics and engineering applications.