Question

Each of the lettered points at the corners of the cube in Fig. Q27.12 represents a positive charge $q$moving with a velocity of magnitude $v$in the direction indicated. The region in the figure is in a uniform magnetic field , parallel to the x-axis and directed toward the right. Which charges experience a force due to $\stackrel{\rightharpoonup }{B}$? What is the direction of the force on each charge?

Short answer

$a$ experiences a force in the negative z direction

$b$ experiences a force in the positive y direction$d$experiences a force in the negative y direction

$e$experience a force 45 degrees below the negative z axis in the yz plane

Step-by-step solution

particle a

Use the right hand rule.

index finger in the y direction. Middle finger in the x direction, thus our thumb points in the negative z direciton. Therefore particle $a$feels a force in the negative z direction.

particle b

Again we use the right hand rule. Index finger in the z direction. Middle finger in the x direction, thus our thumb points in the y direciton. Therefore particle $b$feels a force in the y direction.

particle c

Here the particles direction is antiparallel to the magnetic field, so our particle feels no force.

particle d

This is a slightly more difficult application of the right hand rule. We break our force into components to solve this. We have $d_x$and $d_{-z}$. Note that $d_x$ is parallel to the magnetic field and feels no force, so we only have to consider the other component. Index finger in the negative z direction. Middle finger in the x direction, thus our thumb points in the negative y direciton. Therefore particle feels a force in the negative y direction.

particle e

This is similar to particle $d$, but now we have to consider both components. Here we have a velocity component in the negative z direction and a component in the y direction shown below.

Thus $v_y$feels a force in the negative z direction, and $v_z$feels a force in the negative y direction. Our net force is the sum of these two components. as shown below.