Two long, straight, parallel wires are 1.00 m apart (Fig. 𝐏 2 8 . 5 9 ). The wire on the left c

Two long, straight, parallel wires are 1.00 m apart (Fig. 𝐏...

Key Concepts

Superposition Principle

The superposition principle states that when multiple magnetic fields are present, the net magnetic field at any point is the vector sum of the individual fields produced by each current source. This concept allows for the analysis of complex configurations by adding the effects of each source together, taking into consideration both the magnitude and the direction of each field.

Vector Addition of Magnetic Fields

Since magnetic fields are vector quantities, their combination requires careful vector addition. This involves breaking each magnetic field into its components along a chosen coordinate system and then summing these components to find the net magnetic field. Understanding vector addition is crucial when the fields from different sources are oriented in different directions.

Magnetic Field of a Current-Carrying Conductor

A long, straight wire carrying a steady current produces a magnetic field that encircles the wire. The magnitude of this field can be calculated using the formula B = (??I)/(2?r), where ?? is the permeability of free space, I is the current, and r is the distance from the wire. This relationship is fundamental for determining the individual contributions of each wire to the overall magnetic field at any point in space.

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 flow, the curl of the fingers shows the direction of the magnetic field lines. This rule is essential for establishing the orientation of the fields produced by multiple wires.

Magnetic Interaction between Parallel Currents

The magnetic fields generated by two parallel wires interact with each other. The force between the wires—whether attractive or repulsive—depends on the direction of the currents relative to each other. This concept explains phenomena such as the cancellation or reinforcement of magnetic fields at certain points in space, based on the direction and magnitude of the individual currents.

Transcript

00:02 Hi, in the given problem there are two parallel current carrying conductors.

00:06 The first conductor is carrying a current i1, given to be 6 .00 ampere and that is directed into the plane of paper.

00:19 Now there is another conductor also kept parallel to it, but the magnitude of current carried by this conductor and its direction both are missing now there is a point on the line joining these two conductors the point p which is at a distance of 0 .50 meter from the second conductor the distance between the two conductors is given as 1 .00 meter and it is given that the magnetic field at point p, the net magnetic field, is 0.

01:02 Now using right hand thumb rule, the direction of magnetic field at point p due to the first current carrying conductor, means we can say to be b at p due to 1 as this direction of the current is downward in the plane of paper.

01:41 This is b at p due to 1.

01:49 So in order to get the net magnetic field to be 0, the magnetic field due to the current in the second conductor means b p 2 should be upward in the plane of paper.

02:11 So for this b p 2 to be upward the current i2 in the second.

02:21 Conductor should be coming out perpendicular to the plane of paper.

02:39 So this is one of the answer.

02:41 Means now we know the direction of this current.

02:44 Now in order to get the magnitude of this current, we use the concept b at p due to 1 should be equal to b at p due to 2 for which the expression for the magnetic field at any point around our current carrying conductor using biotrent and severed's law it becomes mu not by 4 pi into 2 i1 by r for current i1 the distance will be 1 .00 meter plus 0 .50 meter so it will come out to be 1 .50 meter is equal to mu not by 4 pi into 2 i2 by the distance is just 0 .50 meter so 3 .5 meter so canceling the similar terms from both the sides.

03:35 Now we get this expression to be equal to.

03:43 And here this current i1 is given as 6 amp.

03:51 So plugging in this value as 6 ampere divided by 1 .5 is equal to i2, which is missing, divided by 0 .5.

04:02 So finally, the magnitude of that current i2 here comes out to be 2 .00 ampere, which is the answer for the first part of this problem.

04:14 Now in the second part of the problem, we have to find magnetic field at another point q here.

04:21 This is the point q having a distance of 0 .50 meter from this first current carrying conductor towards the left.

04:31 So the magnetic field at b at point q, means.

04:37 Means bq will be equal to the addition of two magnetic fields...

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