Ring of charge (half positive, half negative)

[jtran06]

Homework Statement

Two half-rings of charge of opposite polarity are brought together at the origin (so that the rings create a full circle against the y- and z-axis. Each half-ring has a charge of magnitude Q and radius a. Derive the net electric field at point P, located on the +x axis a distance d from the centre of the two half-rings.

Homework Equations

Not sure.

The Attempt at a Solution

I really have no idea how to work this out from what we learned in lecture; "electric field of a ring of charge".. Help, please!

 

[haruspex]

As start, can you figure out the direction of the field there? Let the line of join of the semicircles be the y-axis, say. Consider separately the components of the field in the x, y and z directions. Can you see any that must cancel by symmetry?

 

[jtran06]

I've worked it out conceptually to see that only those in the z direction do not cancel out but I'm at lost on how to retrieve that mathematically.

 

[vela]

Explain to us how to find the field due to a ring of charge then — you know, what you went over in lecture. If you understand what was done there, you should be able to make a dent in this problem.

 

[Nickie]

As a scientist, it is important to approach problems with a systematic and analytical mindset. In this case, we can start by breaking down the problem into smaller, more manageable parts.

First, let's consider the electric field created by each half-ring individually. Since the half-rings have opposite polarities, we can use the superposition principle to find the net electric field. This principle states that the total electric field at a point is the vector sum of the individual electric fields created by each charge.

Next, we can use the equation for the electric field of a ring of charge to calculate the electric field at point P due to each half-ring. This equation is given by E = kQx / (x^2 + a^2)^(3/2), where k is the Coulomb's constant, Q is the charge of the ring, x is the distance from the center of the ring to the point, and a is the radius of the ring.

Now, we can use vector addition to find the total electric field at point P. Since the two half-rings are symmetrically placed along the y- and z-axes, the electric fields due to each half-ring will have equal magnitudes but opposite directions. Therefore, the net electric field at point P will be zero along the y- and z-axes.

Finally, we can use the Pythagorean theorem to find the magnitude of the electric field at point P along the x-axis. This will give us the final answer to the problem.

In summary, by breaking down the problem into smaller parts and using the equations and principles we have learned in lecture, we can derive the net electric field at point P due to the two half-rings of charge. This approach not only helps us solve the problem at hand, but also strengthens our understanding of the underlying concepts and principles.