The drawing shows a wire composed of three segments, AB, BC,...

The drawing shows a wire composed of three segments, AB, BC, and CD. There is a current of $I=2.8 \mathrm{A}$ in the wire. There is also a magnetic field $\overrightarrow{\mathrm{B}}$ (magnitude $=0.26 \mathrm{T}$ ) that is the same everywhere and points in the direction of the $+z$ axis. The lengths of the wire segments are $L_{\mathrm{AB}}=1.1 \mathrm{m}, L_{\mathrm{BC}}=0.55 \mathrm{m},$ and $L_{\mathrm{CD}}=0.55 \mathrm{m}$ . Find the magnitude of the force that acts on each segment.

Key Concepts

Magnetic Force on a Current-Carrying Wire

This concept explains that a conductor carrying an electric current experiences a force when placed in a magnetic field. The force is quantified by the Lorentz force equation, F = I (L × B), where I is the current, L is the length (and direction) of the segment of the conductor, and B is the magnetic field. The magnitude of the force is given by F = I L B sin(?), emphasizing that the force depends on the sine of the angle between the direction of the current and the magnetic field.

Vector Cross Product

The cross product is a mathematical operation between two vectors that results in a new vector perpendicular to the plane defined by the original vectors. In the context of magnetic forces, the cross product between the length vector (representing the current’s direction and magnitude) and the magnetic field vector determines not only the magnitude but also the direction of the force acting on a segment of the wire.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the resultant vector in a cross product. When applied to a current-carrying conductor in a magnetic field, it helps in identifying the direction in which the magnetic force acts. By pointing the fingers of the right hand in the direction of the current and curling them toward the direction of the magnetic field, the thumb then points in the direction of the force.

Transcript

00:02 Okay, this is an interesting problem we have here.

00:05 So this is the setup that we're given.

00:08 We have a wire that starts here, point a, goes to point b, goes to point c, and then goes up to point d.

00:16 This is a three -dimensional grid.

00:21 We're also given the magnetic field that points in the positive z direction.

00:24 So it's in the same direction as this segment c to d.

00:28 And that points perpendicular to the plane created by the x -axis and the y -axis here z points up from that axis you can think of x and y as your paper and then z points straight up out of the page this is three -dimensional so now what we want to find is the force on each segment all right, right off the bat, we can rule out the force from the segment from c to d here because it's in the same direction as b.

01:07 And we have an equation here for force.

01:14 The magnetic force is equal to the current times the length of the current's flowing through, in this case the wire, times the magnetic field present, now the important part is this is sine of theta.

01:43 When the magnetic field is in the same direction as the current, the traveling current, we have sine of zeta.

01:50 Rather, sine of theta is equal to zero, or we'll have theta is zero, and sine of zero is zero.

01:58 So for the line from c to d, we have no angle difference.

02:03 They are in the same direction or parallel, which means that theta is equal to zero.

02:09 So we don't have to worry about that case.

02:13 We can write it right here just so we don't forget.

02:16 Actually, this is c to d.

02:17 We have the segment from c to d is equal to zero.

02:24 The force.

02:27 Zero noons.

02:30 Okay.

02:32 Rule that out.