Calculating Net Electric Field for a Charged Arc

[Jrlinton]

Homework Statement

Homework Equations

L=pi*r/2
λ=q/L
Enet=∑λ(2sinθ)*k

The Attempt at a Solution

So first to calculate L for each arc
L1=pi(.107m)/2
L2=pi(.214m)/2
L3=pi(.321m)/2
and lambda:
λ1=3.3E-6/L1=2E-5
λ2=-12E-6/L2=-3.9E-5
λ3=29.7E-6/L3=5.9E-5
So using θ=pi/2 and rearranging the equation found above with the distributive property i come up with:
k(2sin(π/2)*(2E-5-3.9E-5+5.9E-5)
=508551 N/C
This was incorrect and I am unsure of my mistake(s)

 

[TSny]

Enet=∑λ(2sinθ)*k

This equation is incorrect. You can see that it doesn't have the correct dimensions for an electric field.
(When you use the correct formula, you will see that the electric fields from each arc have an interesting relation.)

 

[Cutter Ketch]

Homework Equations

L=pi*r/2
λ=q/L
Enet=∑λ(2sinθ)*k

I agree with TSny that the last equation is incorrect. I wonder if you were ever given an equation describing the electric field at the center of curvature of an arc. That seems too random. Do you think that perhaps you are supposed to integrate to get the net field of each arc? Has your class been doing similar integral problems?

 

[Jrlinton]

We haven't done much integral problems besides calculating the number of excess electrons in a rod and things of that sort. I guess I am just without a formula for the net field and the center of an arc.

 

[Jrlinton]

I suppose if i were to integrate it I could do so by setting the limits of integration so that the difference is θ but then I am unsure of what to include in the integration. As they are my only known information i have to assume that the radius, linear charge density and constant k is included. And then I would just add the integrations for each arc to find the net field.

 

[gneill]

Hint: Concentrate on finding the field at the radius center of a quarter circle arc. Redraw it to take best advantage of symmetry:

You can use angular charge density for the arc since it's natural to integrate over the enclosed angle.

 

[Jrlinton]

So:
kλ/r∫cos(θ)dθ with the limits of integration being 0 to π/2 to get the x component of the field that it is asking for and to get the y component it would be identical but with the integration being ∫sin(θ)dθ?

 

[gneill]

Use symmetry! You should be able to see that one of the components is perfectly canceled by symmetrically placed charge elements (if you orient the arc as I depicted).

 

[Jrlinton]

Right, so the vertical component is absent...

 

[gneill]

Right, so the vertical component is absent...

Yes, for the shown orientation of the arc. You should be able to express the charge density in angular terms knowing the total angle enclosed by the arc and the total charge.