How Does Charge Affect Boundary Conditions in an Electric Field?

In summary, the potential V for a conducting ball of radius R and total charge Q placed in a homogenous electric field E can be found by first solving the problem for Q=0 and then adding the potential of a uniformly charged sphere (Q/r) to the general solution. Boundary conditions need to be taken into account, and for a conducting ball, the charge Q will only be on the surface of the ball.
  • #1
neworder1
66
0
A conducting ball of radius R and total charge Q is placed in a homogenous electric field E. Find the potential V everywhere.

Some attention needs to be paid to boundary conditions. I know how to calculate this when there is no charge, i.e. Q=0. Then I put V=0 on the surface and V=Ez at infinity, and using these two conditions I can obtain the coefficients in the general solution. My problem is - why these boundary conditions don't work when there is nonzero charge? I don't how to turn the total charge into some kind of boundary condition for V. My reasoning is - I can choose V freely on the surface, since V is determined only up to a constant, and at infinity there is only homogenous field E, so the condition at infinity is the same as before. So the calculations are the same, and so is the result. Where do I take the charge Q into account, then?
 
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  • #2
I can't follow you completely, but since the ball is conducting, the charge Q will be on the surface of the ball only, right?
 
  • #3
neworder1 said:
A conducting ball of radius R and total charge Q is placed in a homogenous electric field E. Find the potential V everywhere.

Some attention needs to be paid to boundary conditions. I know how to calculate this when there is no charge, i.e. Q=0. Then I put V=0 on the surface and V=Ez at infinity, and using these two conditions I can obtain the coefficients in the general solution. My problem is - why these boundary conditions don't work when there is nonzero charge? I don't how to turn the total charge into some kind of boundary condition for V. My reasoning is - I can choose V freely on the surface, since V is determined only up to a constant, and at infinity there is only homogenous field E, so the condition at infinity is the same as before. So the calculations are the same, and so is the result. Where do I take the charge Q into account, then?
For a charge Q, first solve the Q=0 problem as you say.
Then just add the potential of a uniformly charged sphere (Q/r).
 

1. How does an electric field affect a ball?

When a ball is placed in an electric field, it will experience a force due to the interaction between the electric field and the charged particles within the ball. The direction and magnitude of this force will depend on the strength and direction of the electric field, as well as the charge of the ball.

2. What happens when a ball with a positive charge is placed in an electric field?

A positively charged ball placed in an electric field will experience a force in the direction of the electric field. This is because positive charges are attracted to negative charges, and the electric field provides a path for the positive charges to move towards the negative charges.

3. How does the shape or size of the ball affect its movement in an electric field?

The shape and size of the ball will not affect its movement in an electric field. As long as the ball has a charge, it will experience a force in the direction of the electric field. However, the mass of the ball will affect how quickly it accelerates in response to the force.

4. Can a ball move in an electric field without being charged?

Yes, a ball can move in an electric field without being charged. This is because the electric field can induce a temporary charge on the ball, causing it to experience a force in the direction of the electric field. This phenomenon is known as electrostatic induction.

5. How can the movement of a ball in an electric field be controlled?

The movement of a ball in an electric field can be controlled by changing the strength or direction of the electric field. This can be achieved by adjusting the voltage or position of the electric field source. Additionally, the charge of the ball can also be changed to alter its movement in the electric field.

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