Question

Magnets exert forces on other magnets even though they are separated by some distance. Usually the force on a magnet (or piece of magnetized matter) is pictured asthe interaction of that magnet with the magnetic field at its location (the field being generated by other magnets or currents). More fundamentally, theforce arises from the interaction of individual moving charges within a magnet with the local magnetic field. This force is written ,where $F_vec$ is the force, is the individual charge (which can be negative), $v_vec$ is its velocity, and $B_vec$ is the local magnetic field.

This force is nonintuitive, as it involves the vector product (or cross product) of the vectors $v_vec$ and $B_vec$ . In the following questions we assume that the coordinate system being used has theconventional arrangement of the axes, such that it satisfies , where $x_unit$ , $y_unit$ , and $z_unit$ are the unit vectors along the respective axes.

$32652_a2.jpg$ Let's go through the right-hand rule. Starting with the generic vector cross-product equation $vec{A}=vec{B}timesvec{C}$ point your forefinger of your right hand inthe direction of $B_vec$ , and point your middle finger in thedirection of $C_vec$ . Your thumb will then be pointing in thedirection of $A_vec$ .Part AConsider the specific example of a positive charge moving in the+x direction with the local magnetic field in the +y direction. In which direction is the magnetic force acting on the particle?Express your answer using unit vectors (e.g., $x_unit$ - $y_unit$ ). (Recall that $x_unit$ is written x_unit.)Direction of $F_mag_vec$ =Part BNow consider the example of a positive charge moving in the+x direction with the local magnetic field in the +z direction. In which direction is the magnetic force acting on the particle?Express your answer using unit vectors.Direction of $F_mag_vec$ =Part CNow consider the example of a positive charge moving inthe xy plane with velocity (i.e., with magnitude at angle with respect to the x axis). If the localmagnetic field is in the +z direction, what is the direction of the magnetic force acting on the particle?Express the direction of the force in terms of , as a linear combination of unit vectors, $x_unit$ , $y_unit$ , and $z_unit$ .Direction of $F_mag_vec$ =Part DFirst find the magnitude of the force on apositive charge in the case that the velocity $v_vec$ (of magnitude ) and the magnetic field $B_vec$ (of magnitude ) are perpendicular.Express your answer in terms of , , , and other quantities given in the problem statement.F=|q|vBPart ENow consider the example of a positive charge moving in the -z direction with speed with the local magnetic field of magnitude inthe +z direction. Find , the magnitudeof the magnetic force acting on the particle.Express your answer in terms of , , , and other quantities given in the problem statement.F=0 $32652_b.jpg$ Part FNow consider the case in which the positive charge is moving in the yz plane with a speed at an angle withthe z axis as shownin figure above (with the magnetic field still in the +z direction with magnitude). Find the magnetic force $F_vec$ on the charge.Express the magnetic force in terms of given variables like , , , , and unit vectors. $F_vec$ = ?£ex.øi5¡gx.øi5

Answer 1

the answer to part F is qvBsin(θ)x_unit

Answer 2

a.) = k_hat B.) = - j_hat C.) = -cos(theta)j_hat+ sin(theta)k_hat D.) = q*v*B E.) = 0, a charged particle with a velocity that is opposite the magneticfield has no work done on it. F.) = qvBsin(theta) i_hat

Answer 3

the answer to part C is -cos(theta)j_hat + sin(theta)i_hat