The long, straight wire AB shown in Fig. P28.64 carries a current of 14.0 A. The rectangular loo

step 1 Okay, this is question 28 .64. We have one long wire up here with current that I'll call I1, tha

The long, straight wire AB shown in Fig. P28.64 carries a...

Key Concepts

Magnetic Field of a Current-Carrying Conductor

A long, straight wire generates a magnetic field that encircles the wire with a direction given by the right-hand rule. The magnitude of this field decreases inversely with the distance from the wire according to the formula B = ??I/(2?r), where ?? is the permeability of free space, I is the current through the wire, and r is the distance from the wire. This relationship is derived from the Biot–Savart law and Ampère's law and is fundamental in understanding how currents produce magnetic fields.

Lorentz Force on a Current-Carrying Conductor

A segment of a current-carrying conductor placed in a magnetic field experiences a force given by F = I (l × B), where F is the force vector, I is the current, l is the length vector indicating the direction and magnitude of the current element, and B is the magnetic field. The direction of this force is determined by the right-hand rule, and its magnitude depends on the sine of the angle between the current direction and the magnetic field.

Superposition of Magnetic Forces

When analyzing systems with multiple current-carrying conductors or loops in a magnetic field, the net force on the system is obtained by vectorially adding the forces acting on each segment. Since different segments may experience different magnitudes and directions of force due to variations in the magnetic field, the overall force is the cumulative result of these differential contributions.

Net Force on a Current Loop

In a current loop placed in a spatially varying magnetic field, each segment of the loop experiences a force according to the Lorentz force law. Due to the geometry of the loop and the non-uniform magnetic field, forces on different segments may not completely cancel each other out, resulting in a net force. Calculating this net force involves considering both the magnitude and direction of the forces on the various segments of the loop.

Transcript

00:01 Okay, this is question 28 .64.

00:03 We have one long wire up here with current that i'll call i1, that has a current of 14 amps.

00:11 And then we have this wire loop down here with current i2, which i'll call, i'll call this current i2.

00:19 And i2 is 5 amps.

00:21 And we want to know what the force is on the wire, on this loop of the wire, caused by this long wire up on the top.

00:33 So to do that, we're going to need to use our equation for the force on two segments of wire.

00:41 And we want to know the actual absolute force.

00:43 So let's multiply both sides here by l so that we are just solving for the force directly.

00:52 Now, since these wires in this loop are separated by just different distances, we're going to need to consider each side of the loop separately.

01:02 So let's first think about the these two horizontal lines over the two vertical parts of the loop.

01:11 If you think about it, the magnetic field exerted on both of these from this wire loop is going to be the same.

01:18 But the current in either of these is in opposite directions.

01:22 So when we add up the sum of the force on this wire and the sum of the force on this wire, we're going to get two values that are equal in magnitude but opposite in direction.

01:35 So one of these is going to be negative and one will be positive.

01:37 But that means that they're essentially just going to cancel out because they're equal in magnitude but opposite in direction.

01:45 So we can just ignore these two vertical parts of the loop because they're not going to contribute anything to the force.

01:54 They're just going to cancel.

01:55 So now let's consider, we're going to have to just add the force.

01:59 On the top to the force on the bottom.

02:03 And let's just think about this conceptually.

02:06 We have the force on the top, we have two wires that are going in the same direction, with currents going in the same direction.

02:13 And when we have two currents going in the same direction, they will attract each other.

02:18 So this is going to be an attractive force.

02:21 But on this bottom wire, the currents are going in opposite directions from the original wire.

02:26 So that means that these ones are going to repel...

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