(II) Three long parallel wires are 3.8 cm from one another. (Looking along them, they are at thr

step 1 First we will calculate net force on wire M. Here wire N exerts a refulsive force on wire M as t

(II) Three long parallel wires are 3.8 cm from one another....

Key Concepts

Geometrical Symmetry in Current Configurations

When current-carrying wires are arranged in a symmetric geometrical configuration, such as being at the corners of an equilateral triangle, symmetry can be used to simplify the analysis of the magnetic forces. The relative positions and distances between the wires ensure that the magnetic fields and forces have predictable magnitudes and directions, aiding in the calculation of the net force on each wire.

Superposition Principle in Magnetism

In systems with multiple current-carrying wires, the net magnetic field and the resulting forces are obtained by vectorially adding the contributions from each wire individually. This superposition principle allows one to analyze the effect of each wire independently and then combine the effects to determine the overall force on a given wire.

Right-Hand Rule for Magnetic Fields and Forces

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying wire as well as the direction of the force acting on a current when placed in an external magnetic field. By pointing the thumb in the direction of the current, the curled fingers indicate the circumferential direction of the magnetic field. Similarly, for forces, the cross product in the Lorentz force law (F = I(L x B)) defines the direction of the force on the wire.

Magnetic Field Around a Current-Carrying Wire

A current-carrying wire produces a magnetic field in the space surrounding it. The magnitude of this field at a distance r from the wire is given by B = ??I/(2?r), where ?? is the permeability of free space. This relationship is a direct consequence of Ampère’s law and is crucial for understanding how currents interact magnetically with their environment.

Magnetic Force Between Parallel Currents

Two parallel wires carrying currents experience a magnetic force due to the interaction of the magnetic field of one wire with the current in the other. The force per unit length is given by f = ??I?I?/(2?r), where I? and I? are the currents in the wires and r is the separation between them. The force is attractive when the currents flow in the same direction and repulsive when they flow in opposite directions, showcasing the inherent relationship between current direction and electromagnetic interaction.

Transcript

00:02 First we will calculate net force on wire m.

00:12 Here wire n exerts a refulsive force on wire m as the direction of currents are opposite in both wires.

00:25 Also the wire p exerts a repulsive force on wire m as the direction of currents are opposite in wire m and wire p.

00:39 Interior angles of a peculator triangle are equal to 60 degree so by using the idea of alternate and vertical angles the angle between forces and horizontal line are also equal to 60 degree so first we will calculate the magnitude of force exerted by wire and on wire so by using the relation force on wire m by n equal to mu not times current through wire m into current through wire n into length of wire m divided by 2 pi into distance between the two wires putting different values here mu not is 4 pi times 10 to the power minus 7 tesla meter per ampere into current through m is at ampere and current through n is also at ampere into length of wire m divided by 2 pi distance between wire m and n that is actually equal to the side length of triangle which is a 3 .8 centimeter that is 3 .8 times 10 to the power minus 2 meter this gives us the force that is a magnitude of force exerted by wire n on wire m equal to 3 .366 times 10 to the power minus 4 times length of wire m and the units are here newton per meter similarly the force exerted by wire p on m is equal to mu not times current through wire m into current through wire p into length of wire m divided by 2 pi distance between wire p and wire m again putting different values here mu not is equal to 4 pi times 10 to the power minus 7 tesla meter per ampere current through m is at ampere current through wire p is at ampere into length of wire m divided by 2 pi distance between wire m and p is 3 .8 centimeter that is 3 .8 centimeter that is 3 .8 times 10 to the power minus 2 meter this gives us the magnitude of force exerted by wire p on wire m equal to 3 .666 times 10 to the power minus 4 times length of wire m and units are newton per meter now we will calculate x component of net force on wire m by using the relation f that is x component of net force on m equal to x component of force exerted by wire n on wire m here magnitude of this component is f mn into cosine of 60 degree minus x component of the force exerted by p on wire m m is f mp cosine of 60 degree as the x component of these forces are equal in magnitude but they are opposite in direction so this gives us the x component of net force on wire m equal to zero now the the y component of net force on m is equal to the magnitude of y component of force exerted by wire n on wire m which is f m sine of 60 degree plus the y component of the force exerted by p on m now by putting different values here for magnitude of force exerted by wire n on em is 3 .666 3 .3 .3 .3 .366 x6 times 10 to the power minus 4 times length of wire m units or newton per meter into sine of 60 degree plus magnitude of force exerted by wire p on m is also 3 .3 6x times 10 to the power minus 4 into length of wire m newton per meter into sign of 60 degree this gives us this gives us the y component of net force on wire m equal to 5 point and times 10 to the power minus 4 minus 4 into length of wire m newton per meter now we want to calculate magnitude of net force by using the relation magnitude of net force on wire n equal to square root of suquer of x component of net force plus square of y component of net force as x component on as x component of net force on yrm is equal to 0 plus y component of net force is 5 .8 times 10 to power minus 4 into length of wire m so this gives us the magnitude of net force on m equal to 5 .8 times 10 to the power minus 4 times length of wire m newton per meter or we can write this as magnitude of net force divided by length of wire m equal to 5 .8 times 10 to the power minus 4 newton per meter so this is the magnitude of force on unit length of wire m due to other wires...

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