Consider the conductor and current in Example 27.8 , but now...

Consider the conductor and current in Example $27.8 ,$ but now let the magnetic field be parallel to the $x$ -axis. (a) What are the magnitude and direction of the total magnetic force on the conductor? (b) In Example 27.8 , the total force is the same as if we replaced the semicircle with a straight segment along the $x$ -axis. Is that still true when the magnetic field is in this different direction? Can you explain why, or why not?

Consider the conductor and current in Example $27.8 ,$ but now let the magnetic field be parallel to the $x$ -axis. (a) What are the
magnitude and direction of the total magnetic force on the conductor? (b) In Example 27.8 , the total force is the same as if we
replaced the semicircle with a straight segment along the $x$ -axis. Is that still true when the magnetic field is in this different direction?
Can you explain why, or why not?

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept explains that a current-carrying conductor placed in a magnetic field experiences a force given by the cross product F = I (L × B). The force’s magnitude depends on the current, the length of the conductor within the field, and the sine of the angle between the current direction and the magnetic field vector, while its direction is perpendicular to both the current element and the magnetic field.

Lorentz Force Law

The Lorentz force law governs the behavior of moving charges in electromagnetic fields. In the context of a conductor, it is applied to the drift velocity of charges, resulting in a force on the conductor equivalent to summing the individual forces on each moving charge, thereby reinforcing the use of the cross product for computing the total magnetic force.

Vector Integration Along a Conductor

When analyzing forces on conductors with non-straight geometries, integration becomes necessary. This concept involves summing the differential force contributions along the entire path of the conductor to determine the net force, accounting for variations in the conductor’s orientation relative to the magnetic field.

Effect of Magnetic Field Orientation

The direction of the magnetic field relative to the conductor’s geometry can have a significant impact on the net force. Since the magnetic force is determined by the sine of the angle between the current segment and the magnetic field, changing the field orientation can alter both the magnitude and the direction of the resultant force, potentially leading to scenarios where replacing a curved segment with a straight one does not yield the same net force.

Transcript

00:01 Okay, we have a problem here dealing with the magnetic force on a current.

00:05 In this problem, we have a conductor that's sort of set up like this.

00:10 We have a circular current connected to this flat current in the x direction.

00:16 And we also have a current that's coming in from out of the page.

00:24 So a current i -n, which also has the same length l as the straight segment over here.

00:30 What we want to find out is what is the total force on this due to this magnetic field b that's pointed in the x direction.

00:42 So that is part a here.

00:45 So we're going to do this piece by piece.

00:47 So i'm going to call this straight segment here, just part one.

00:53 I'm going to call the circular part two.

00:56 And then this thing coming from out of the page, part three.

01:02 Okay, so first off, this straight segment pointed in the x.

01:05 Direction.

01:08 What we'll use here is this equation.

01:11 So force on a current due to a magnetic field is the current times the length crossed with the magnetic field.

01:23 Now here we see that the current and the magnetic field are both pointed in the x direction, right? one's pointing in negative x, the other one's pointed in the positive x.

01:34 But since they're both in the x direction, this is going to be zero since the cross product between two vectors pointed in the same direction is zero.

01:47 So now we can move on to part two.

01:51 So here, now we have to consider the fact that this vector, this l vector, is going to be changing over the entire conductor...

Please give Ace some feedback