# Lorentz Force Law to determine the magnitude and direction of an electric field

#### c_m

1. The problem statement, all variables and given/known data
Ok here it goes...

A stream of electrons, each with speed u = 5.9 x 10^6 m s^-1 and travelling along the x-axis in the positive x-direction enters a region pervaded by a uniform magnitic field B. The electrons the describe a circle with raidus R in the horizontal xy-plane, circulating anticlockwise as viewed from above.

With varius questions i have calculated the.....
1)the direction of B is in the -Z direction
2)Magnitude of B = 1.7 x 10^-3 T

Now for the part i have become brain dead on...

Two large, parallel, metal plates 0.20m apart and placed symetrically on either side of the incoming electron beam, are now used to apply a uniform electric field in the region where the electrons were circulating.

This electric field is such that the electron beam is no longer deflected but continues straight on in the original positive x-direction.

My question is:
1)Starting from the Lorentz force Law and using newtons first law, How do i determine the direction of the electric field (E)?

2. Relevant equations
Lorentz Force Law in a uniform field
F = q[E + u x B

B=magnetic field
q=charge on electron
u=speed of electron
E = electric field

3. The attempt at a solution
Now i assume that i need to find the magnitude of the electric field, so when i rearranged Lorentz Law this is what i ended up with:

qE = B x q x u

But when i input my values going by the units which are:
T x C x m s^-1 this gives me Newtons
where i know the units of electric charge should be N C^-1

So obviously i have gone very wrong somewhere probably with my magnitude expression??

Please help, if you need any more details, let me know and i will try and supply them.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

#### Kruum

Welcome to PF!

qE = B x q x u, here lies your problem. You're trying to determine the magnitude of the electric field, right? You should correct it a bit.

#### c_m

yea thats what im trying to, any chance you could give me a little push in the right direction as to where i have gone wrong? because i figured that where i made a mistake but i cant see how to correct it?

#### Kruum

I suggest you try to solve $$F=q(E+ (\vec {v} \times \vec {B}))$$ for $$E$$ again. That's where your problem is.

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#### c_m

ok, i have given it one last attempt before my brain gives up, i now have,

E = qu x B / q x F

am i right? or closer to been right?

#### Kruum

ok, i have given it one last attempt before my brain gives up, i now have,

E = qu x B / q x F

am i right? or closer to been right?
That's correct! But what can you say about the force? Note that the electrons are going straight at a constant speed.

#### c_m

Wow! i suprised myself there!

wouldnt the force be perpendicular to u and b?

#### Kruum

wouldnt the force be perpendicular to u and b?
Yes it would, but what about it's magnitude? You'll need it to determine E.

#### c_m

Well isnt the magnitude of F, F=ma?

.....wish my brain would wake up!

#### Kruum

Well isnt the magnitude of F, F=ma?
Yep, and since the electron has a constant speed, isn't the magnitude 0? And with a fresh set of eyes I spy a little mistake in your equation. It should be $$E=F- \frac{qvB}{q}$$. Now you should be able to solve the magnitude od electric field from the equation and use the information given in the problem to determine the direction.

#### c_m

Iv got it now! thankyou very much you were a great help!

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