A conducting sphere has a net charge of +Q . Points A and B are on opposite sides of the surface of the sphere. If a point charge +q is moved from point A to point B in a straight line through the center of the sphere, the work done to move the charge is WC . If a point charge +q is moved from point A to point B along the surface of the sphere, the work done is WS . How do Wc and Ws compare?
Wc= Ws because while as the electric field inside a conductor is zero, there is no force that acts on the charge as it passes through the conductor resulting in zero work using the common definition of work as Fd = W. On the other hand there is a force that exists on the outside of the sphere as that is where the charges of the conducting sphere are located however those are parallel to the area vector (perpendicular to the surface) and thus when the charge moves along the sphere's surface it is effectively moving along an equipotential line and perpendicular to the electric force, thus it experiences no work either.
Well, you see, when you move a charge through the center of a conducting sphere, it's like taking a shortcut. It's the most efficient way to move around because you don't have to face any obstacles or bumps along the way.
So, the work done to move the charge through the center of the sphere, WC, is actually going to be smaller compared to moving it along the surface, WS. It's kind of like taking the express lane instead of navigating through traffic. Less effort, more efficiency!
So, in short, WC will be less than WS because moving through the center of the conducting sphere is the smoothest and easiest route for our little charge buddy.
To compare the work done to move a point charge +q from point A to point B in a straight line through the center of a conducting sphere, versus moving the charge along the surface of the sphere, we need to consider the electric field and potential.
When the charge is moved in a straight line through the center of the sphere, the electric field at every point on the path is directed along the path. This means that the electric field and the direction of displacement are always parallel, resulting in zero work done by the electric field. Therefore, the work done to move the charge from A to B in a straight line through the center of the sphere, Wc, is zero.
On the other hand, when the charge is moved along the surface of the conducting sphere, the electric field at each point is not parallel to the displacement. The electric field lines are perpendicular to the surface of the sphere at every point, while the displacement is tangent to the surface. This means that the electric field and displacement are perpendicular to each other, resulting in non-zero work done by the electric field.
Since work done is given by the equation: work done = force × displacement × cos(angle between force and displacement), and the angle between the electric field and displacement is 90 degrees along the surface of the sphere, the work done to move the charge from A to B along the surface, WS, is nonzero.
In conclusion, the work done to move the charge from A to B in a straight line through the center of the sphere, Wc, is zero, while the work done to move the charge along the surface of the sphere, WS, is non-zero.
To compare the work done when moving the point charge +q from point A to point B, we need to consider the difference in the paths taken.
1. Work Done via a Straight Line (WC):
When the charge +q is moved in a straight line through the center of the conducting sphere, the electric field inside the sphere is zero. This is because the conducting sphere redistributes its charges to counteract the electric field created by the charge +q. As a result, no work is done to move the charge when it follows this path. Therefore, WC = 0.
2. Work Done Along the Surface (WS):
When the charge +q is moved along the surface of the conducting sphere from point A to point B, the electric field inside the sphere is no longer zero. The electric field at any point over the surface of a uniformly charged sphere is given by the equation E = kQ/R², where k is Coulomb's constant, Q is the charge on the sphere, and R is the radius of the sphere. As the charge +q is being moved along the surface, it experiences a force due to the electric field, resulting in work being done.
To calculate the work done, we need to consider that the displacement along the surface of the sphere is tangential to the electric field lines. Since the force and displacement are perpendicular to each other, the work done is given by the equation: WS = F * s, where F is the magnitude of the force and s is the displacement between points A and B along the curved surface.
The magnitude of the force can be calculated using the equation F = q * E, where q is the charge and E is the magnitude of the electric field. Since the electric field is constant on the surface of a uniformly charged conducting sphere, F = q * (kQ/R²).
The displacement s along the surface of the sphere is the circumference of a great circle on the sphere. This can be calculated using the equation s = 2πR.
Combining these equations, we can express the work done as WS = q * (kQ/R²) * 2πR = 2πkqQ.
Comparing WC and WS:
Since WC = 0 and WS = 2πkqQ, we can conclude that the work done when moving the charge +q along the surface of the conducting sphere (WS) is non-zero, while no work is done when moving the charge in a straight line through the center of the sphere (WC). In other words, WS is greater than WC.