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

In Fig. 29-57, four long straight wires are perpendicular to the page, and their cross sections form a square of edge length $a=20 \mathrm{~cm} .$ The currents are out of the page in wires 1 and 4 and into the page in wires 2 and $3,$ and each wire carries 20 A. In unitvector notation, what is the net magnetic field at the square's center?

Key Concepts

Magnetic Field from a Long Straight Conductor

A long straight current-carrying conductor produces a magnetic field that encircles the wire. The magnitude of the field at a distance r is given by the formula B = ??I/(2?r), where ?? is the permeability of free space and I is the current. This expression is derived from Ampère’s Law and describes how the field diminishes with distance from the wire.

Principle of Superposition

The principle of superposition states that when multiple sources of magnetic fields are present, the net magnetic field at any point is the vector sum of the individual fields produced by each source. This concept allows for complex magnetic field configurations to be analyzed by breaking them down into simpler, individual contributions.

Right-Hand Rule

The right-hand rule is a method used to determine the direction of the magnetic field around a current-carrying conductor. By pointing the thumb of the right hand in the direction of the current, the way the fingers curl naturally indicates the circular direction of the magnetic field lines. This rule is essential for predicting the field orientation in various electromagnetic systems.

Transcript

00:02 Hi, in the given problem four parallel current carrying conductors of infinite lengths are arranged in the form of the square means the four wires are arranged perpendicular to the plane of paper and these wires are behaving like to be arranged at the vertices of this square.

00:43 Here this is wire one, this is wire two, then wire three, and then wire four.

00:53 All the four wires are kept perpendicular to the plane of paper such that here this is x -axis, this is y -axis, this is a square having a side a where the side a is having a magnitude of 20 centimeter or we can say this is 0 .2 meter all these four wires are carrying same current the current having same magnitude 20 ampere each but the directions the wire 1 and 4 are carrying current coming out of the perpendicularly to the plane of paper and wire 2 and 3 are carrying the same currents but getting into the plane of paper now if we look at these diagonals the length of diagonals as we know in a square the diagonals are given as side root 2 means it will be a root 2 or we can say this is 0 .2 into root 2 meter.

02:12 So if we consider this point o at the center of the square then the distance of center of the square from all these current carrying wires will be given by half of the diagonal means it will come out to be 0 .1 square root of 2 which is 1 .414 meter or it can be given as 0 .14 meter now we will find the magnetic fields at this point o whose directions will be found directions of all the magnetic fields at the center of square are found using right hand thumb rule and using that rule the directions of magnetic field first of all if you look for the direction of magnetic field due to the wire at 4 at means the wire 4 this will be here the magnetic field due to the wire number 4 this is b4 and and then due to the wire at 1 and that will be sorry for this wire 4 the magnetic field direction of magnet field due to wire 4 that will be here using right hand thumb rule this is b4 then due to wire 1 due to wire 1 it will be here then due to wire 2 it will be again here this is b2 magnetic field at the center due to wire at point 2 then finally due to 3 this is here this is b3 so now the magnitudes of all these magnetic fields will be same.

05:02 So b1 is equal to b2 is equal to b3 is equal to b4 and that will be equal to mu not upon 4 pi into 2 i by r...

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