Electric Field and Potential of a Line Charge on a Semi-Circular Sector Boat?

[Pouyan]

Homework Statement

A line charge has the total charge Q evenly distributed over a circle boat with radius a and sector 2β, placed according to the figure

Find the Electric field E and the potential V in the origin.

Homework Equations

I know for this case that E(r) = (1/4πε) ∫ (λ(r')/R²)R dl' , R is unit a vector

λ(r') = Q/(2βa) R = - a *(cos(φ)x + sin(φ)y) dl'=a*dφ R²= a^2

V(r) = (1/4πε) ∫ (λ(r')/R) dl'

The Attempt at a Solution

The electric field is not so hard to calculate I know
E(r) =Q/(2βa)(1/4πε) ∫ ((-a *(cos(φ)x + sin(φ)y))/a³) adφ
which is : (Q/8πεa²β) * [ -sinφx + cosφy]

Now if I want to calculate the potential I know :
V(r) = ∫ E*dl where dl = a*dφ*R = -a*dφ*(cos(φ)x + sin(φ)y)
But I see (Q/8πεa²β) * [ -sinφx + cosφy] * (-a*dφ*(cos(φ)x + sin(φ)y)) = 0!

And the real solution is :

V(r) = (1/4πε) ∫ (λ(r')/R) dl' = (1/4πε) * (Q/2βa) ∫(adφ/a) and for a boat circle from π-β to π+β

Or :

What is actually wrong ?! If I do the same algorithm for a Sphere, it gives me a correct answer but why can't I do the same thing here through ∫ E*dl ?

 

[mfb]

which is : (Q/8πεa²β) * [ -sinφx + cosφy].

There is a step missing to evaluate the electric field.

For the potential, there is an easier way, without using the electric field. Otherwise you would have to calculate the electric field everywhere along your integration path and that gets much more complicated.

 

[Pouyan]

There is a step missing to evaluate the electric field.

For the potential, there is an easier way, without using the electric field. Otherwise you would have to calculate the electric field everywhere along your integration path and that gets much more complicated.

Ok the whole solution is :

Why can't we just integrate E*dl from this solution for E ?!

 

[mfb]

That looks right.

 

[Pouyan]

That looks right.

But is this wrong to think that E =(Q/8πεa²β) * [ -sinφx + cosφy] ?! If we just want to find V through E *dl ?!
I know that there is an easier way. Just using formula. But when I will find potential for a solid sphere it seems much more easier to use ∫E*dl. First we find E and then we integrate over that E and find V. But here I see we can not do exact the same thing ... why ?!

 

[mfb]

What is φ in that formula? How can the electric field depend on it?
The electric field has a zero y component due to the symmetry of the setup.

But here I see we can not do exact the same thing ... why ?!

You can do it, but it is much more work because there is no simple expression for E at other places.

 

[Pouyan]

What is φ in that formula? How can the electric field depend on it?
The electric field has a zero y component due to the symmetry of the setup.
You can do it, but it is much more work because there is no simple expression for E at other places.

It's for polar coordinate.

 

[Pouyan]

You can do it, but it is much more work because there is no simple expression for E at other places.

OK and because of that the easiest way is : V(r) = (1/4πε) ∫ (λ(r')/R) dl' ...

 

[vela]

But is this wrong to think that E =(Q/8πεa²β) * [ -sinφx + cosφy] ?! If we just want to find V through E *dl ?!

Yes, it's wrong. You're integrating over $\phi$ with definite limits, so that variable shouldn't appear in the final result.

I know that there is an easier way. Just using formula. But when I will find potential for a solid sphere it seems much more easier to use ∫E*dl. First we find E and then we integrate over that E and find V. But here I see we can not do exact the same thing ... why ?!

You're calculating the electric field at one point, the origin. To find the potential by integration, you need to find $\vec{E}$ at all points along whatever path you choose from infinity to the origin. In other words, what you're proposing to do is calculate
$$-\int_C \vec{E}(0)\cdot d\vec{r}$$ instead of
$$-\int_C \vec{E}(r)\cdot d\vec{r}$$ where $C$ is some path from infinity to the origin.

 

[mfb]

It's for polar coordinate.

The polar coordinate of what?
The electric field can only depend on parameters of the setup ($\beta$, a and Q). It cannot depend on intermediate parameters introduced in the calculation. If someone gives you values for these three things you have to be able to calculate the electric field in V/m. What would you plug in for $\phi$?

OK and because of that the easiest way is : V(r) = (1/4πε) ∫ (λ(r')/R) dl' ...

Right.

 

[Pouyan]

Yes, it's wrong. You're integrating over $\phi$ with definite limits, so that variable shouldn't appear in the final result.You're calculating the electric field at one point, the origin. To find the potential by integration, you need to find $\vec{E}$ at all points along whatever path you choose from infinity to the origin. In other words, what you're proposing to do is calculate
$$-\int_C \vec{E}(0)\cdot d\vec{r}$$ instead of
$$-\int_C \vec{E}(r)\cdot d\vec{r}$$ where $C$ is some path from infinity to the origin.

OK than you !

What do I see; it's not easy to find E(r) for this case...