Use the linear charge density  and length element , where each point is  from the point . The potential is therefore

Use the linear charge density  and length element . The distance from each point on the ring to the point on the axis is . Lastly, integrate over  from  to  to obtain

Calculate the potential due to one side of the bar, and then multiply this by  to get the total potential from all four sides. Orient the bar along the x-axis such that its endpoints are at , and use the linear charge density . The potential is therefore

According to the shell theorem, the total electric field at the center point of a charged spherical shell is always zero. At this point, any electric field lines will result in symmetry, canceling each other out and creating a net field of zero at that point.

The original charge Q is negative, as indicated by the direction of the electric field lines in the diagram. A negative point charge of any value placed at point A will cause both charges to feel a mutually repelling force on each other. Therefore, the second charge will be repelled by the original negative charge with a force pointing radially along a line connecting the center of Q and the point A, resulting in free movement toward C. Additionally, point B is on a line of equipotential to point A; negative point charges will move from regions of lower electric potential to higher electric potential (against the direction of the electric field lines), so point B is not a viable answer.

To calculate the magnetic force of a single charge, we use , where  is the charge of the proton,  is its velocity,  is the uniform magnetic field.

Since this magnetic force causes the proton to travel in a circular path, we set this magnetic force equation equal to the centripetal force equation.

 is the mass of the proton and  is the radius of the circular path. Solve for .

Using the values given in the question, we can solve for the radius.

The net magnetic flux (or net field flowing in and out) through any closed surface must always be zero. This is because magnetic field lines have no starting or ending points, so any field line going into the surface must also come out. In other words, "there are no magnetic monopoles."

Relevant equations:

Set the magnetic force equal to the centripetal force, since the magnetic force is directed towards the center of the particle's circular path and centripetal force is defined as the net force towards the center of a circular path.

Rearrange to isolate  the velocity:

Determine the distance, , traveled in one revolution, which is the circumference of a circle of radius :

Plug this distance and velocity into , to solve for the period :

The correct answer is zero. To calculate the force of a magnetic field on a moving charged particle, we use the cross product. We know that if the magnetic field is parallel to the velocity vector of the particle, then the force produced is zero.

Because our magnetic field in this case is going in the same direction as the velocity of the particle, we know that the magnetic force on the particle is zero.

The answer is that the wires will move from each other. Using our right hand rule, we know that the magnetic fields produced by each wire are in the same direction, as long as their currents oppose. Using the right hand rule again to determine the direction of the force exerted on each wire by the magnetic field with which they are interacting yields a force in the direction away from the other wire for each wire.