What Is the Electromagnetic Force on an Electron in Given Fields?

[Roodles01]

Homework Statement

An electron in an electric field E = 3.0x10⁷ e_x NC^-1 and a magnetic field B = 3.0 e_z T has velocity v = 1.0x10⁷ (2e_x - e_y) ms^-1.

Calculate electromagnetic force on the electron.

Homework Equations

F = q (E + v X B)
where q = e^- = -1.6x10^-19 C

The Attempt at a Solution

I have attached a working out, but have difficulty with just one aspect.

Whilst working out the matrix vXB I have a negative sign where I think a positive sign should be.
I have attached the working as a pic.
Could someone show me why I am correct or incorrect.
Thank you.

 

[Roodles01]

Ah! Is this to do with e_x X e_y = -e_y?
They're orthogonal?

 

[ap123]

The negative sign comes from the definition of the cross product.
v x B = e_x(...) - e_y(...) + e_z(...)

 

[Roodles01]

Again, I have to go "Doh!"
Thank you.

 

[Nickie]

I am happy to provide a response to your question about the Lorentz force acting on an electron in an electromagnetic field.

First, let's review the Lorentz force equation: F = q (E + v x B), where q is the charge of the particle (in this case, the electron), E is the electric field, v is the velocity of the particle, and B is the magnetic field.

In your attempt at a solution, you correctly identified the values for E, v, and B, and correctly substituted them into the equation. However, you seem to be having trouble with the cross product (v x B).

The cross product of two vectors, v and B, is always perpendicular to both vectors. In this case, since v is in the x-y plane and B is in the z direction, the result of the cross product (v x B) will be in the -x direction. This means that the components of the cross product vector will be negative in the x direction, but positive in the y direction.

To see this more clearly, let's write out the cross product in terms of its components:

(v x B) = (vx, vy, vz) x (0, 0, B) = (0, 0, vx*B)

Since vx = 1.0x107 and B = 3.0, the x-component of the cross product (vx*B) will be -3.0x107. Similarly, the y-component of the cross product will be +3.0x107, since vy = -1.0x107 and B = 3.0.

Therefore, the overall result of the cross product (v x B) will be ( -3.0x107 ex + 3.0x107 ey + 0 ez).

Substituting this into the Lorentz force equation, we get:

F = q (E + ( -3.0x107 ex + 3.0x107 ey + 0 ez)) = -q (3.0x107 ex - 3.0x107 ey) = -(-1.6x10-19 C) (3.0x107 ex - 3.0x107 ey) = 4.8x10-12 ex - 4.8x10-12 ey N

So, the electromagnetic force on the electron is 4.8x