Calculating charge on objects in arrangement?

Question

Consider four masses arranged as shown. Each is 1 cm from the center of the cross. Mass A has a charge +1 uc , mass B has a charge of -1 uc, and mass C has a charge of +2 uc. At the center the electric field points 30 degrees from the vertical axis as shown.

I know that $E = \frac{kq}{r^2}$, and that the distance between each mass is the square root of two centimeters.

However, how do I determine the charge of mass D, the unknown, given what I have? Do I need to use that angle? If so, how?

Thank you!

Answer 1

Use that $$\vec E=\frac{\vec F}{q}$$ and the superposition principle.

Notably that for some constant c we have

$$\vec F_A=c(0,-1) \quad \vec F_B=c(1,0) \quad \vec F_C=c(0,2) \quad \vec F_D=c(x,0)$$

thus

$$\sum F_i=c(1+x,1)$$

and for the given condition we must have

$$\frac{1+x}{1}=\tan 30° = \frac{\sqrt 3}3 \implies x= \frac{\sqrt 3}3 -1$$

Answer 2

All charges have distance $1$ cm to centre.

Consider a small positive test charge at the centre :

$\vec E:$

In $+ Y$ direction (positive, upward): $+1.$

In $+ X$ direction (to the right):

$(1/3)√3=\tan(30°).$

$B$ has a charge of $-1$, hence you need a charge of $-1+(1/3)√3$ at D.