A magnetic field of 0.080 T is in the y -direction. The velocity of wire segment S has a magnit

A magnetic field of 0.080 T is in the y -direction. The...

A magnetic field of $0.080 \mathrm{~T}$ is in the $y$ -direction. The velocity of wire segment $S$ has a magnitude of $78 \mathrm{~m} / \mathrm{s}$ and components of $18 \mathrm{~m} / \mathrm{s}$ in the $x$ -direction, $24 \mathrm{~m} / \mathrm{s}$ in the $y$ -direction, and $72 \mathrm{~m} / \mathrm{s}$ in the $z$ -direction. The segment has length $0.50 \mathrm{~m}$ and is parallel to the $z$ -axis as it moves. (a) Find the motional emf induced between the ends of the segment. (b) What would the motional emf be if the wire segment was parallel to the $y$ -axis?

Key Concepts

Motional EMF

Motional electromotive force (EMF) is the voltage induced in a conductor when it moves through a magnetic field. It arises because the magnetic field exerts a force on the mobile charges within the conductor, causing them to separate, and the induced EMF is calculated from the relative velocity of the conductor, the magnetic field strength, and the effective length of the conductor perpendicular to the field.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field and is given by the cross product of the velocity vector and the magnetic field vector. This force is responsible for the separation of charges along the conductor, leading directly to the generation of motional EMF.

Vector Cross Product

The vector cross product is a fundamental mathematical operation used in physics to determine a vector that is perpendicular to two given vectors. In the context of motional EMF, it is used to calculate the effective direction and magnitude of the induced voltage by combining the velocity of the conductor and the magnetic field.

Component Decomposition

Component decomposition refers to breaking down a vector into its perpendicular components along the coordinate axes. This technique is essential when analyzing the contributions of different velocity components to the induced EMF, helping to ascertain which components effectively contribute based on their orientation relative to the magnetic field.

Conductor Orientation

Conductor orientation is crucial because the induced EMF depends on the alignment of the conductor relative to the magnetic field and its direction of motion. Only the portion of the motion that is perpendicular to the magnetic field contributes to the electromotive force, so the orientation of the conductor can significantly affect the magnitude of the induced voltage.

Transcript

00:02 Hi, in the given problem, there is a magnetic field having a magnitude of 0 .080 tesla, and this is directed along y axis.

00:25 So if we show it in the figure, here this is y -axis, this is x -axis, this is x -axis, and this is z -axis.

00:44 Given that the straight conductor is lying along z -axis.

00:53 Here, this is a straight conductor, and this is the magnetic field.

01:05 The component of velocity of this is straight conductor are such that along x -axis, this is vx, along y -axis, this is v -y, and along z -axis, this is v -z.

01:18 Having the magnitudes as vx is equal to 18 meter per second, vy is equal to 24 meter per second, and vz is equal to 72 meter per second.

01:40 Now, in the first part of the problem, we have to find motional emf induced across the ends of the conductor.

01:50 Now, as we know, the motionally mf induced across the ends of a conductor which is allowed to move in a magnetic field is given as the product of magnetic field with the velocity and with the length of the conductor.

02:10 And there is a condition in order to induce this motional emf and that condition says the magnetic field, the velocity vector and the length of the conductor.

02:28 All these three should be perpendicular to each other as length of the conductor is along x xx here.

02:39 So if we look for the component of velocity along z -axis, this is parallel to length of the conductor.

02:48 So no emf will be induced due to vz component of velocity along z axis and this y component component of velocity along y x is this is along magnetic field.

03:14 So no emf will be induced due to this vy also...

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