A circular loop has radius R and carries current I2 in a clockwise direction (Fig. P28.68). The

A circular loop has radius R and carries current I2 in a...

Key Concepts

Magnetic Field from a Circular Loop

The magnetic field at the center of a circular current loop can be calculated using the Biot-Savart law. This field is given by a formula that depends on the permeability of free space, the current flowing through the loop, and the radius of the loop. The direction of this field follows from the right?hand rule applied to the current’s direction in the loop. This concept is key for understanding how current in a closed loop contributes to local magnetic fields.

Magnetic Field from a Long Straight Wire

A long straight wire carrying a current produces a magnetic field whose magnitude is determined by Ampère’s law, resulting in an inverse relationship with the distance from the wire. The magnetic field circulates around the wire according to the right?hand rule. This concept is fundamental for calculating the magnetic influence of linear segments of current-carrying conductors.

Superposition Principle in Magnetism

In magnetic phenomena, the net magnetic field at any point is the vector sum of the individual fields produced by each current source. This principle allows the analysis of complex configurations by calculating the contribution from each element separately and then determining the condition for cancellation or reinforcement of the fields, such as achieving a zero net field at a specific point.

Right-Hand Rule in Magnetism

The right-hand rule is a mnemonic for determining the direction of the magnetic field produced by a current-carrying conductor or loop. By orienting the right hand with the thumb in the direction of the current flow, the curled fingers indicate the direction of the magnetic field circles around the conductor. This rule is vital for predicting the direction of magnetic fields in various geometries and is used to ensure proper vector summation during analysis.

Transcript

00:01 Okay, we're looking at question 2872, where we have a circular loop with current i2 going clockwise.

00:09 And then we also have a wire down here with a current, but we don't know what direction the current is going in.

00:15 But we want the magnetic field at the center of this loop, so this point right here, to be zero.

00:22 So what that means is the magnetic field from this loop is going to have to cancel out the magnetic field from, this wire.

00:32 So first, let's figure out the magnitude of the magnetic field coming from this wire.

00:39 So here's the expression for the magnetic field from an infinite line of current.

00:46 So here we have this, we're talking about the, this wire down here.

00:53 So this is i -1.

00:54 And the distance that we're talking about here is going to be d.

00:58 So this is a wire down here.

01:00 So this is a one.

01:00 And the distance that we're talking about here is a are let's call this b1, the magnetic field coming from this current down here.

01:07 And again, we don't know which direction this is coming.

01:10 This is going to exert on yet.

01:12 So now for b2, we have a loop of current, and we're looking at the magnetic field right in the center.

01:20 So the expression for the magnetic field from a loop of current is going to be we have mu not times i 2 here this is the numerator and then in the denominator is 2 times the radius and the radius here is r so this is going to be the magnetic field coming from this loop and we do know the direction that this is going to be in because we can use the right -hand rule so using the right -hand rule curling our fingers around where the current loop is going that means that it's going to point into the board or into the page here.

02:05 So we know that this one is going to be into the page, so we know that this is going to need to be out of the page in order to cancel it out.

02:15 So let's just figure out the direction of the current here right away, since we know before we get the magnitude.

02:23 In order to get a current, a magnetic field out of the page from a current here, we know that this current is going to need to go to the right.

02:36 Because if we have a current going to the right, again, if we point our thumb in the direction of the magnetic field of the current here, that will give us a magnetic field coming out of the page at this point...

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