Calculating the Electric field at a point due to a ring of charge

[Yosty22]

I don't know if this is the correct section. It is not exactly a homework problem, but here it is:

If I were given a circle of charge with radius r and were asked to find the electric field due to this circle of charge at the center of the circle, would it be valid to do the following:

Since I know the radius of the circle of charge, could I imagine the circle of charge to be a line of charge, and the point in question be r away. That is, find the circumference of the circle, as if I were stretching out the circle into a straight line of length equal to the circumference of the circle. Then, I could calculate the electric field due to the line of charge at a distance equal to the radius of the circle.

Is this valid to do? I am sure there would be an easier way to solve this problem, but just out of curiosity, would this work?

 

[CAF123]

Is this valid to do? I am sure there would be an easier way to solve this problem, but just out of curiosity, would this work?

I don't think it would be valid. In the case of a circular source of charge with the centre of the circle at point P, all the charge elements are at a distance r away. If instead you keep P fixed, but put a horizontal line of charge a distance r away, not all the charge elements (or points on that line) will be at a distance r from P.

 

[Yosty22]

Ok, that makes sense. What about if it was a ring with constant current through it and you were looking for the b field at the center. Would it work in this case?

 

[CAF123]

Ok, that makes sense. What about if it was a ring with constant current through it and you were looking for the b field at the center. Would it work in this case?

No, for the same reason, the current elements are not all the same distance from the point of interest. Using the Biot-Savart law, you can derive explicit expressions for the B field at the centre of the loop from a circular current flow and that from a wire.

 

[CaptainHammer]

Just apply Gauss's law and you're ready to go.

The scenario you described corresponds to a closed surface. Therefore, you need the area of the circle, not it's perimeter.

 

[Chestermiller]

Shouldn't the electric field at the center of the circle be zero as a result of the symmetry of the geometry. Which direction would you think the electric field vector at the center of the circle would be pointing?

Chet