Tho long wires carry 10 A curin the directions shown in...

Tho long wires carry 10 A curin the directions shown in Figure P24.7. One wire is $10 \mathrm{cm}$ above the other. What are the direction and magnitude of the magnetic field at a point halfway between them?

Key Concepts

Magnetic Field of a Long Straight Conductor

This concept explains that a long, straight current-carrying wire produces a magnetic field in the space surrounding it. The magnitude of this magnetic field follows the relation B = ??I/(2?r), where I is the current, r is the radial distance from the wire, and ?? is the permeability of free space. This relation is derived from Ampere’s law and forms a fundamental tool for calculating the magnetic fields in various configurations involving straight conductors.

Right-Hand Rule

The right-hand rule is a method used to determine the direction of the magnetic field produced by a current-carrying conductor. When the thumb of the right hand points in the direction of the current, the curl of the fingers indicates the circular direction of the magnetic field lines around the conductor. This technique is essential for visualizing and predicting the behavior of magnetic fields in relation to the orientation of currents.

Superposition Principle in Magnetic Fields

The superposition principle states that when multiple magnetic fields are present, the total magnetic field at any given point is the vector sum of the individual fields produced by each current source. This principle allows for the analysis of complex magnetic field arrangements by breaking them down into simpler components whose effects can be calculated independently and then combined to obtain the overall magnetic field.

Transcript

00:01 In the given problem, there are the two straight conductors.

00:06 This is the lower one carrying a current eye, having a magnitude of 10 ampere towards right.

00:19 And above it, another wire carrying the current eye, having the same magnitude, of 10 ampere in downward direction and the gap between these two wires is given as 10 centimeter so the gap between these two wires is 10 centimeter we have to find the net magnetic field in magnitude in and in direction at a point half way between them.

01:06 So the point will be at a distance of r by 2 means 5 cm or we can say 5 into 10 -1 minus 2 meter from both the wires.

01:23 Now if we look for the directions of magnetic fields due to the individual wires for this horizontal wire, the direction of magnetic field at a point above it will be in a downward direction, means like this.

01:48 This is the direction of magnetic field due to this horizontal wire.

01:55 Similarly, the direction of magnetic field at the same point due to this vertical wire using right hand thumb rule again will be towards right.

02:11 So this is the magnetic field due to the vertical wire.

02:17 So the net magnetic field will be found by vector sum of these two magnetic fields.

02:29 Now using bayard and severus law, magnetic field due to a straight conductor of infinite length, at any point, distance r from the wire is given as mu -not by 4 pi into 2 i by r as the current is same in both the wires and distance of observation point is same is also same from both the wires.

02:56 So the two magnetic fields, b1 and b2, will be same in magnitude...

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