Question

A straight wire carries a 10.0-A current. ABCD is a rectangle with point D in the middle
of a 1.10-mm segment of the wire and point C in the wire. Find the magnitude and
direction of the magnetic field due to this segment at (a) point A; (b) point B; (c) point C.

Answer 1

Concepts and reason

The concept required to solve the given question is Biot – Savart Law.

First, calculate the magnetic field produced by the wire that carries a current using the Biot – Savart Law. It gives the expression for magnetic field due to a current carrying wire.

Later, use the direction of the fingers, the direction of the magnetic field can be calculated. Repeat the same method for the other distances also.

Fundamentals

Biot – Savart Law:

Biot – Savart Law gives the relation between the magnetic field and the current carrying elements. Consider a current carrying wire. Each infinitesimal element of current gives a contribution to the magnetic field at a point P that is perpendicular to the current element. The calculation of a magnetic field due to the current element involves a vector product.

The mathematical expression for Biot – Savart law is given by,

Here, is the magnetic field at the point r , I is the current carried by the wire, is the infinitesimal length of the conductor carrying current, is the unit vector that specifies the direction of the distance r from the current to the point, is the permeability of free space.

The direction of magnetic field can be calculated by using the right-hand thumb rule which deals with placing the right-hand thumb in the direction of the current and curling the fingers around the conductor. Then, the curled fingers give the direction of the magnetic field.

The distance between two points can be calculated using Pythagoras theorem. It is applicable to right angled triangles.

Here, c is the hypotenuse, and b and a are the legs of the right-angled triangle.

(a)

Magnitude and direction of magnetic field at the point A:

Consider the expression for a magnetic field using Biot – Savart law,

Substitute for , 10 A for I, for dl , for and 5 cm for r.

The direction of the magnetic field at the point A is out of the page.

(b)

The magnitude and direction of magnetic field at the point B:

The distance between the points D and B,

Substitute 5 cm for AD and 14 cm for AB.

The angle between the current carrying wire and the line BD,

Substitute 5 cm for CB and 0.148 m for DB.

The magnetic field at point B,

Substitute for , 10 A for I, for dl , for and for r.

The direction of the magnetic field at point B moves out from the page.

(c)

Magnitude and direction of magnetic field at the point C:

The angle between the points D and C is,

Thus, the magnitude of the magnetic field at point C,

$dB=0T $

Ans: Part a

The magnitude of the magnetic field at point A is and the direction moves out from the page.