Magnetic Forces

To solve this problem we simply use the right hand rule. First we construct a three dimensional axis, as shown below. Then we point our thumb in the positive x direction, our index finger in the positive y direction, and we find that our middle finger points in the positive z direction, implying that this is exactly the direction of the force on the particle.

Finding the magnitude of the cross product is easy: it is simply v ₁ v ₂sinθ = (10)(10)(.5) = 50 . The direction of the cross product, however, takes a little thought. Since we are computing v ₁×v ₂ , think of v ₁ as a velocity vector, and v ₂ as a magnetic field vector. Using the right hand rule, then, we find that the cross product of the two points in the positive z direction. Notice from this problem that cross products are not communative: the direction of v ₁×v ₂ is the opposite of that of v ₂×v ₁ . This problem should help with the complicated directions of fields, velocities, and forces.

To solve the problem we use the equation:

So we must find the vector sum of the electric force and the magnetic force. The electric force is easy: it is simply qE = 10q in the positive x direction. To find the magnetic force, we must use the right hand rule (again), and find that the force on the particle must act in the negative x direction. Thus we must now find the magnitude of the force. Since v and B are perpendicular, we do not need to compute a cross product, and the equation simplifies to F _B = = = 10q . Since this force acts in the negative x direction, it exactly cancels the electric force on the particle. Thus, even though both an electric field and a magnetic field act on the particle, it experiences no net force.

In this case the magnetic field produces the centripetal force required to move the particle in uniform circular motion. We know that, since v is perpendicular to B , the magnitude of the magnetic force is simply F _B = . We also know that any centripetal force has magnitude F _c = . Since the magnetic force is the only one acting in this situation, we may relate the two quantities:

Analyzing our answer we see that stronger fields cause particles to move in smaller circles.