Electric Field due to two circular line charges

[emmett92k]

Is the integral of $dq$ not the integral of Rd\theta?

 

[blue_leaf77]

$dq$ is equal to $\lambda R d\theta$.

 

[emmett92k]

Ah so I end up with V_1 = \frac{Q}{4{\pi}r_1\epsilon}, V_2 = \frac{-Q}{4{\pi}r_2\epsilon}

This give a total V of:

V = \frac{Q}{4\pi\epsilon}\Bigg[\frac{1}{r_1} - \frac{1}{r_2}\Bigg]

Is this correct?

 

[blue_leaf77]

Yes that's the correct answer.

 

[emmett92k]

Ok thanks for all the help by the way.

For part (c) I know its partial differentiation but what will I differentiate with respect to? Will I do r_1 and r_2 separately?

 

[blue_leaf77]

Ok thanks for all the help by the way.

You are welcome.

For part (c) I know its partial differentiation but what will I differentiate with respect to? Will I do r_1 and r_2 separately?

Part c) is a bit tricky because as you said we need to apply nabla operator $\nabla$ to the potential at arbitrary point. But what we have at hand is the potential at points along the axis only. The trick is to make use of the symmetry of the system. So, first start by writing $V(x,y,z)$ as the potential at any point in space, not just on the axis. We want to calculate
$$\mathbf{E}(x=0,y=0,z) = -\nabla V(x,y,z) \big|_{x=0,y=0} = - \bigg( \hat{x}\frac{\partial V(x,y,z)}{\partial x}|_{x=0,y=0} + \hat{y}\frac{\partial V(x,y,z)}{\partial y}|_{x=0,y=0} + \hat{z}\frac{\partial V(x,y,z)}{\partial z}|_{x=0,y=0} \bigg)$$
Where the requirement x=0 and y=0 indicates that the observation point is on the axis which is taken to be the z axis. Now apply some symmetry argument to find the value of the first two terms without really calculating it.

 

[emmett92k]

Well the first two terms would be equal to zero. How do I differentiate the $z$ part if there is no $z$ variable?

 

[blue_leaf77]

That's right but I hope you know why they must be zero. To reveal the position of z, you need to express $r_1$ in term of the corresponding ring radius and the distance from the ring center to the observation point, which is z. Do the same for the other ring.

 

[emmett92k]

Ok I'm starting to follow now. The r values will be the distance of the hypotenuse if I draw a triangle. So:

r_1 = \sqrt{a^2 + z^2}, r_2 = \sqrt{b^2 + z^2}

So I sub them and differentiate to get:

E = \frac{Q}{4\pi\epsilon}\Bigg[\frac{2z}{(a^2+z^2)^\frac{3}{2}} - \frac{2z}{(b^2+z^2)^\frac{3}{2}}\Bigg]

This simplifies to:

E = \frac{Qz}{2\pi\epsilon}\Bigg[\frac{1}{(a^2+z^2)^\frac{3}{2}} - \frac{1}{(b^2+z^2)^\frac{3}{2}}\Bigg]

This is different to my E in part (a) however.

 

[blue_leaf77]

You are being careless in calculating the derivatives.

 

[emmett92k]

Sorry I've found the mistake, the 2 multiplies by a half and cancels.

As regards part (d) do I treat it as a dipole and use the formula:

E(r,\theta) = \frac{qd}{4{\pi}{\epsilon}r^3}(cos{\theta}\hat{r}+sin\dot{\theta}\hat{\theta})

 

[blue_leaf77]

I don't know if that works. But anyway, task d) is not so difficult compared to task a). The way you do it is the same as we did in a), only that now there is only one ring but with upper half positively charged and lower half negatively charged. Then make use of the symmetry property to deduce which vector components of the E field should vanish.

 

[emmett92k]

So does that mean I split the circle into two semi-circles and use the same method as (a) but change \lambda = \frac{Q}{2{\pi}R} to \lambda = \frac{Q}{{\pi}R} because its half the circumference?

 

[blue_leaf77]

So does that mean I split the circle into two semi-circles and use the same method as (a) but change \lambda = \frac{Q}{2{\pi}R} to \lambda = \frac{Q}{{\pi}R} because its half the circumference?

Yes but that will come later.
For this problem it's easier if we work in spherical coordinate and translate the observation point to the origin, so that the ring will lie on a plane located at $z$. I will start with the E field due to the positive charge first
$$ \mathbf{E}_+ = k\int_0^\pi \frac{\lambda R d\phi}{r}\hat{r}$$
Next, express $\hat{r}$ in its cartesian components.

 

[emmett92k]

For $E_+$ is $\hat{r}$ r(0,a,z)?

 

[blue_leaf77]

For $E_+$ is $\hat{r}$ r(0,a,z)?

Nope, $\hat{r}$ is the unit vector from a particular point on the ring to the origin, which means it's also the same as the inverse of the radial unit vector in spherical coordinate. So you will want to know how to express radial unit vector in spherical coordinate into cartesian components, $\hat{x}$, $\hat{y}$, and $\hat{z}$.
You can find what you are looking for in https://en.wikipedia.org/wiki/Spherical_coordinate_system under "Integration and differentiation in spherical coordinates" section.

 

[emmett92k]

Spherical coordinates are $(r,\phi,\theta)$ and cartesian is $(x,y,z)$.

$r=\sqrt{x^2+y^2+z^2}$, $\theta=tan^{-1}\big(\frac{y}{x}\big)$, $\phi=cos^{-1}\big(\frac{z}{r}\big)$

How do I precede from here?

 

[emmett92k]

Actually is it:

$x=rcos{\theta}sin\phi$
$y=rsin{\theta}cos\phi$
$z=rcos\phi$

 

[blue_leaf77]

Not that one, look at the link I just added in my previous comment. What I meant is the expression of the unit vector $\hat{r}$, not the coordinates themself.

 

[emmett92k]

Did you see I left two comments there? Because my first comment is what is on that page?

Do you mean $\hat{r}=\frac{\vec{r}}{r}$

 

[blue_leaf77]

Look for something like $\hat{r} = \ldots \hat{x} + \ldots \hat{y} + \ldots \hat{z}$.

 

[emmett92k]

Yeah so:

$\hat{r}=cos{\theta}sin\phi\hat{x}+sin{\theta}cos\phi\hat{y}+cos\phi\hat{z}$

 

[blue_leaf77]

No, check again whether you copied it right.

 

[emmett92k]

Sorry don't know how I did that:

$\hat{r}=sin{\theta}cos\phi\hat{x}+sin{\theta}sin\phi\hat{y}+cos\phi\hat{z}$

 

[blue_leaf77]

Now put that into the integral in comment #44 and do the integration, remember the integration variable is $\phi$ while $\theta$ is a constant (I leave to you to express the sine or cosine of $\theta$ in terms of known quantities). You should find that there are only two components surviving the integration. Upon considering (no calculation is needed) the contribution from the other half of the ring, one more component should vanish, and the only remaining components should be enhanched by a factor of two. Good luck.

 

[emmett92k]

Quick question, those the upper limit stay as $\phi$? Thanks for all the help with this question, hopefully it all pays off.

 

[blue_leaf77]

That's a mistake, see again I have corrected it.

 

[emmett92k]

What I end up with is:

$E_+=\frac{Q}{2\pi\epsilon_0} \frac{1}{\sqrt{a^2+z^2}}$

Is this the answer I should've gotten?

 

[blue_leaf77]

Obviously that can't be right answer as the unit doesn't match that of an electric field.