Electric and magnetic fields of a moving charge

[TSny]

"The distance between the two charges are x" was the full sentence. The given result is just the expression I got for E(r,t).
View attachment 323650
When drawing the figure I just realized $$\mathbf{r_2}$$, which connects the negative charge to the field point is not $$\mathbf{r_1}+\mathbf{x}$$. That would have been the case if the field point were to be right above the charges, with both angles being zero. Finding an expression for $$\mathbf{r_2}$$ will be difficult, so I will just leave it as $$\mathbf{r_2}$$. So the electric-and magnetic field lines will look like:
$$\mathbf{E}(\mathbf{r},t) = \frac{q \left(1 - v^2 / c^2 \right) \mathbf{\hat{r_1}}}{4 \pi \epsilon_0 \left( 1 - v^2 \sin^2 (\theta_1 )/c ^2 \right)^{3/2}r_{1}^2} - \frac{q \left(1 - v^2 / c^2 \right) \mathbf{\hat{r_2}}}{4 \pi \epsilon_0 \left( 1 - v^2 \sin^2 (\theta_2 )/c ^2 \right)^{3/2}r_{2}^2}$$
$$\mathbf{B}(\mathbf{r},t) = \frac{q \mu_0 \left(1 - v^2 / c^2 \right) \sin(\theta_1) \mathbf{\hat{\phi}}}{4 \pi \left( 1 - v^2 \sin^2 (\theta_1 )/c ^2 \right)^{3/2}r_{1}^2} + \frac{q \mu_0 \left(1 - v^2 / c^2 \right) \sin(\theta_2) \mathbf{\hat{\phi}}}{4 \pi \left( 1 - v^2 \sin^2 (\theta_2 )/c ^2 \right)^{3/2}r_{2}^2}.$$
I guess.

As a vector equation, you could write $\mathbf{r}_2 = \mathbf{x} + \mathbf{r}_1$, where $\mathbf{x}$ is the vector from $q_2$ to $q_1$. So, $$r_2 = |\mathbf{x} + \mathbf{r}_1| = \sqrt{x^2 + r_1^2 + 2xr_1\sin\theta_1}$$ I'm not able to read your handwriting too well. In your diagram it looks like you are using the symbol $\theta_2$ for the angle of $\mathbf{r}_1$ and $\theta_1$ for the angle of $\mathbf{r}_2$. It could just be my poor eyes not reading your subscripts on the angles very well .

Otherwise, Your expression for the total E field looks good to me. The B field expression also looks good except I believe you dropped a factor of the speed $v$ in the numerators on the right side. Of course, I don't know if this form of writing the results is what is expected by the person who made up the problem.

 

[TSny]

Also I'm a bit confused with what they mean with "at origin" does that mean that the field point is right above the charges or?

Yes, I don't know how to interpret the statement of the problem. When I first read it, I thought the two charges are moving in opposite directions along the same line so that they meet at the origin at $t = 0$ and pass through each other. But I really don't know. I hesitate to try to go further with this problem until we can be clear on the interpretation of the problem statement. I don't want to lead you off in the wrong direction.

 

[TSny]

It's net, this is what I got View attachment 323645

OK. This looks about right for the net E field if the charges pass each other at some finite distance. For non-relativistic speeds, the electric field will be a dipole field as you've drawn. The E-field lines will be modified somewhat at relativistic speeds.

For the B field, you have indicated the fields of the two charges individually. It looks OK. It would not be easy to indicate the net B field from this perspective. Maybe you could sketch the net B field from the perspective of looking from the right so that the + charge is moving toward you and the - charge is moving away from you. So the plane of the page would be the plane passing through the charges and the plane oriented perpendicular to the direction of motion of the particles. The field lines of one of the charges alone would be circular. You could try to sketch the net B-field pattern.

 

[milkism]

As a vector equation, you could write $\mathbf{r}_2 = \mathbf{x} + \mathbf{r}_1$, where $\mathbf{x}$ is the vector from $q_2$ to $q_1$. So, $$r_2 = |\mathbf{x} + \mathbf{r}_1| = \sqrt{x^2 + r_1^2 + 2xr_1\sin\theta_1}$$ I'm not able to read your handwriting too well. In your diagram it looks like you are using the symbol $\theta_2$ for the angle of $\mathbf{r}_1$ and $\theta_1$ for the angle of $\mathbf{r}_2$. It could just be my poor eyes not reading your subscripts on the angles very well .

Otherwise, Your expression for the total E field looks good to me. The B field expression also looks good except I believe you dropped a factor of the speed $v$ in the numerators on the right side. Of course, I don't know if this form of writing the results is what is expected by the person who made up the problem.

Thanks :D, yes theta_2 is for r_2 and theta_1 for r_1, and yep I forgot to add the velocity in the magnetic field lol. I have sent a message to the professor about the interpretation of "at the origin".