In Fig. 29-57, four long straight wires are perpendicular to the page, and their cross sections

In Fig. 29-57, four long straight wires are perpendicular to...

Key Concepts

Vector Representation in Electromagnetism

Using vector notation in electromagnetism allows for precise expression of both the magnitude and direction of physical quantities such as magnetic fields and forces. Representing these quantities as vectors is essential for calculating cross products, like those found in F = I L × B, and for systematically combining contributions from multiple sources.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor as well as the direction of the force acting on a conductor in a magnetic field. By aligning the thumb with the current direction and curling the fingers, one can ascertain the direction in which the magnetic field circulates, thereby aiding in the determination of force directions.

Superposition Principle in Electromagnetism

The superposition principle states that the net magnetic field at any point is the vector sum of the individual fields produced by each current source. This principle is crucial when evaluating the total force on a wire due to multiple sources, as it allows for the independent calculation of each wire's contribution and then summing their vector effects.

Magnetic Force on a Current-Carrying Conductor

The magnetic force on a current-carrying wire placed in an external magnetic field is given by F = I L × B, where I is the current, L is the length vector of the wire, and B is the magnetic field vector. This concept allows us to determine the magnitude and direction of the force acting on conductors in a magnetic environment, which is the basis for analyzing the interactions among wires carrying currents.

Magnetic Field of a Long Straight Current-Carrying Wire

This concept describes the magnetic field generated by a long, straight conductor carrying a steady current. According to the Biot–Savart law or Ampère’s circuital law, the magnetic field circulates around the wire and its magnitude decreases inversely with distance from the wire. This field is central to analyzing interactions between multiple current-carrying wires.

Transcript

00:01 In the given problem, the four parallel current carrying conductors are a range such that they form a square.

00:15 Means they are kept at the vertices of this square, which is having a side a and these conductors this is the conductor number four which is carrying a current coming out of the plane of paper this is conductor one which is also carrying the current coming out of the plane of paper this is conductor number two which is carrying a current getting into the plane of paper and finally conductor number three that is also carrying the current directed into the plane of paper here this is x -axis and this is y -axis so side of the square is given as 13 .5 centimeter which can also be written as 0 .135 meter and the current carried by each conductor is same and that is having a magnitude of 7 .50 ampere.

01:26 Now in this problem we have to find net force experienced by this conductor core.

01:32 Per unit of its length in unit vector notation.

01:38 So first of all, this conductor 4 is carrying the current in the same direction as in conductor number 1.

01:46 So it will be attracted by the conductor 1 means the force acting on 4 due to 1 is attractive in nature.

01:55 So this is we will name it f at 4 due to 1 but at the same time this 4 will be repelled by conductor 3 as the currents are in opposite direction and the force will be having same magnitude but different direction and it will be named as effect 4 due to 3 now if we mark its diagonal also diagonal of the square between its vertices 2 and 4 it will be given as the expression for diagonal is side root 2 means here it will be 0 .135 into root 2 meter and as again the conductor 4 and conductor 2 are carrying currents in opposite direction so the 4 conductor number 4 will be repelled by this conductor 2 also so the force will be acting like this and this is named as f at 4 due to 2 and here this angle will be 45 degree.

03:07 So this f4 -2 may be resolved into two components.

03:10 Here this will be f -4 -2 -cos 45 degree and here this will be along negative y -axis...

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