A rectangular coil of size 20 cm ×10 cm is placed in a horizontal uniform magnetic field of magn

A rectangular coil of size 20 cm ×10 cm is placed in a...

A rectangular coil of size $20 \mathrm{cm} \times 10 \mathrm{cm}$ is placed in a horizontal uniform magnetic field of magnitude $0.050 \mathrm{T}$, as shown in Figure $6.30 .$ A current of 2.0 A flows in the coil in a counter-clockwise direction as shown. (a) Find the force on sections $\mathrm{AB}$, $\mathrm{BC}$, $\mathrm{CD}$ and $\mathrm{DA}$. (b) What is the net force on the coil?

A rectangular coil of size $20 \mathrm{cm} \times 10 \mathrm{cm}$ is placed in a horizontal uniform magnetic field of magnitude $0.050 \mathrm{T}$, as shown in Figure $6.30 .$ A current of 2.0 A flows in the coil in a counter-clockwise direction as shown.
(a) Find the force on sections $\mathrm{AB}$, $\mathrm{BC}$, $\mathrm{CD}$ and $\mathrm{DA}$.
(b) What is the net force on the coil?

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept involves the force experienced by a current-carrying wire when it is placed in a magnetic field. The magnitude and direction of the force are given by the cross product F = I (L × B), where I is the current, L is the length vector of the conductor, and B is the magnetic field vector. This fundamental idea describes how moving charges interact with magnetic fields to produce forces acting perpendicular to both the current direction and the magnetic field.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the force acting on a current-carrying conductor in a magnetic field. By pointing the fingers in the direction of the current and curling them toward the magnetic field, the thumb points in the direction of the force. This rule is essential for visualizing and predicting the vector nature of the magnetic force in different geometrical arrangements.

Uniform Magnetic Field

A uniform magnetic field is one where the magnitude and direction of the magnetic field do not vary with position. In such fields, current-carrying segments experience forces that are consistent in magnitude and predictable in direction, leading to simplifications in the analysis of force distributions and net forces on objects like current loops.

Net Force on a Closed Loop

In a uniform magnetic field, the net force on a closed current-carrying loop is generally zero because the forces on opposing segments cancel out. This cancellation occurs even though individual segments of the loop experience non-zero forces. Understanding this concept is fundamental when analyzing the overall effect of magnetic fields on current loops, as it highlights the difference between distributed forces and net translational force.

Transcript

00:02 In the given problem, there is a uniform magnetic field in horizontal direction shown like this.

00:16 The magnitude of this magnetic field is given as 0 .050 tesla.

00:24 Then there is a rectangular current carrying loop kept in this magnetic field like this.

00:33 This loop is carrying a current in counterclockwise direction like this.

00:53 This loop is named as a, b, c and d.

01:02 The dimensions of this loop are given as here.

01:06 The length of this loop is 20 cm and its width is given as 10 centimeter.

01:18 The current passing through this loop is having a magnitude of 2 .0 ampere.

01:27 Now in the first part of the problem we have to find and the force, magnetic force is experienced by all these four arms of this loop.

01:38 First of all, for the force acting at the arm ab and using the expression for this force in vector form, it is given as il means current element in vector form.

01:52 So for l it will be ab.

01:55 This is iab across the magnetic field in vector form.

02:01 So here it will come out to be fab is equal to i into a b the length of the wire into b here this b is for magnetic field into sine 90 degree as these two are perpendicular to each other.

02:16 So finally plugging in all known values for the current this is 2 .0 amper for length of the wire this is 20 centimeter or 20 by 100 meter multiplied by the magnetic field which is 0 .5...

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