Q. 80

Expert-verifiedFound in: Page 835

Book edition
4th

Author(s)
Randall D. Knight

Pages
1240 pages

ISBN
9780133942651

a. Derive an expression for the magnetic field strength at distance d from the center of a straight wire of finite length l that carries current I.

b. Determine the field strength at the center of a current carrying square loop having sides of length 2R.

c. Compare your answer to part b to the field at the center of a circular loop of diameter 2R. Do so by computing the ratio .

(a) .

(b) .

(c) .

The length of the wire is , the current carrying by the wire is and the distance of the observation point from the center of the wire is .

Let's consider the following diagram.

Let's assume that AB is a straight wire of length and carrying current . Also consider P be any point which is at a distance from the wire where the magnetic field is to be calculated. Also consider an elemental length on the wire which makes an angle with respect to the position vector from the point P.

From the diagram, it can be written that

Also, from the diagram it can be written that,

Equate equation (1) and (2) and simplify to obtain the expression for the elemental length.

From , it can be written that

From Biot-Savart's law, the magnetic field at point P due to the elemental length is given by

Here, is the infinitesimal magnetic field and is the permeability of free space.

Substitute the expression for and from equation (3) and (4) respectively into equation (5) and simplify to obtain the infinitesimal magnetic field.

The formula to calculate the net magnetic field at P is given by

Substitute the expression for from equation (6) into equation (7) and simplify to obtain the required magnetic field.

The required magnetic field is given by .

The square loop has each side length of .

A square loop can be thought of made up of four individual wires carrying the same current.

The formula to calculate the magnetic field at a distance due to a current carrying wire is given by

As the square loop has each side length of , it can be calculated from basic geometry that the distance of the center of the square from the center of any side is .

The net magnetic field at the center of the square is, then, given by

Substitute the expression for from equation (7) and for to obtain the required magnetic field.

The required magnetic field is given by .

The diameter of the circular loop is .

The formula to calculate the magnetic field at the center of a circular loop of diameter is given by

Divide equation (8) by equation (9) and simplify to obtain the required ratio.

The required ratio is .

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