A particle accelerating in both magnetic and electric fields

[joriarty]

Homework Statement

A proton (q = 1.6×10⁻¹⁹ C, m = 1.67×10⁻²⁷ kg) moves at 300 ms⁻¹ along a line midway between the x and z axes. There is a magnetic field of 1T along the x axis and an electric field of 2000 Vm⁻¹ along the y axis. What is the force on the proton and what is its acceleration?

Homework Equations

Scalar product: a·b=abcosθ=a_xb_x +a_yb_y +a_zb_z.

Vector product: a×b = i(a_yb_z −a_zb_y)+j(a_zb_x −a_xb_z)+k(a_xb_y −a_yb_x), |a×b| = absinθ.

Forces on a charged particle in electric and magnetic fields: F = qE + qv × B.

The Attempt at a Solution

First off, my main assumption is that there shouldn't be any relativity involved in this question since v=300m s^-1 is not particularly fast. However when I arrived at the answer I have calculated below, that is quite a lot of acceleration! I didn't end up needing to use the scalar and vector product formulae that were provided on the question sheet, so I fear that I am missing something!

I have drawn a diagram to show how I have interpreted the question, which is attached. The proton is traveling with initial velocity v=300m s^-1 along a line where x=z and y=0 (which I assume is what my lecturer means when he says "along a line midway between the x and z axes).

So the total force acting on the particle F=F_E+F_B where F_E is the force due to the electric field of 2000 Vm⁻¹ and F_B is the force on the proton due to the magnetic field of 1 T.

So F=qE+qvB

But what is v? Because only the component of motion perpendicular to the magnetic field (the velocity along the z-axis) affects the force, v=300sin(Pi/4)=150\sqrt{2} m s^-1.

Using the right hand slap rule for the magnetic field, I can see that the force exerted on the proton by the magnetic field is in the same direction as that exerted by the electric field on a positive particle, that is, straight along the y-axis. Therefore I should be able to just add these two forces.

Therefore F = 1.6×10⁻¹⁹ C × 2000 Vm^-1 + 1.6×10⁻¹⁹ C × 150\sqrt{2} m s^-1 × 1 T = 3.54×10⁻¹⁶ N along the y-axis.

Then using a=F/m = (3.54×10⁻¹⁶ N) / (1.67×10⁻²⁷ kg) = 2.12×10¹¹ m s^-2 in the y-direction.

Does all of this working seem to make sense to you? I can't shake the feeling that I have missed something - why would my lecturer give me the scalar and vector product formulae if I did not have to use them?

 

[ehild]

Your method is all right and you do use the scalar and vector products :).

When you get the velocity components, you take the scalar product of the velocity vector v with the unit vectors. So v_z= v˙k .

The magnetic force is the vector product qvxB= q(v_xi + v_zk) x Bi=qv_zj.

ehild

 

[joriarty]

Ah I see it now! Those operations just seemed to me like "common sense", I guess I didn't analyse my method enough. Typical. Thank you for your clarification :)