Problem 14 - University Physics with Modern Physics

Question

How is it possible that the flux through a closed surface does not depend on where inside the surface the charge is located (that is, the charge can be moved around inside the surface with no effect whatsoever on the flux)? If the charge is moved from just inside to just outside the surface, the flux changes discontinuously to zero, according to Gauss's Law. Does this really happen? Explain.

Step-by-Step Solution

Answer

Answer: According to Gauss's Law, the electric flux through a closed surface depends on the total enclosed charge and not on the individual positions of the charges inside the surface. As long as a charge remains inside the surface, the flux remains the same, even if the charge moves. However, when the charge moves from inside to just outside the surface, the total enclosed charge and electric flux through the closed surface change, potentially becoming zero if no other charges are enclosed within the surface.

Step 1: Understand Gauss's Law

Gauss's Law states that the electric flux through any closed surface is equal to the total electric charge enclosed within that surface divided by the permittivity of free space (ε₀). Mathematically, it can be written as: Φ = Q_enclosed / ε₀ where Φ is the electric flux, Q_enclosed is the total enclosed electric charge, and ε₀ is the permittivity of free space. We will use this law to explain the given scenario.

Step 2: Electric flux and enclosed charge

The electric flux through a closed surface depends on the total enclosed charge, rather than the individual charges' positions inside the surface. As long as the enclosed charge doesn't change the electric flux will remain the same. When we move a charge inside the surface, the total enclosed charge remains the same, hence the flux remains the same according to Gauss's Law.

Step 3: Moving the charge to the surface

As the charge is moved to the surface, it is still considered to be inside the surface, and therefore, the total enclosed charge does not change. Consequently, the electric flux through the closed surface remains the same.

Step 4: Moving the charge outside the surface

When the charge is moved just outside the surface, the enclosed charge inside the surface changes. In this case, the charge is now not enclosed by the surface, and the total enclosed charge decreases by the magnitude of the moved charge. Since the enclosed charge decreases, the electric flux through the closed surface will also change according to Gauss's Law. The flux will decrease and would be zero if there are no other charges enclosed within the surface.

Step 5: Conclusion

In conclusion, Gauss's Law states that the electric flux through a closed surface is proportional to the total enclosed charge. As long as the charge remains inside the surface, the flux will not change, regardless of its position. However, when a charge is moved from inside to just outside the surface, the total enclosed charge and electric flux through the closed surface changes, potentially becoming zero if no other charges are enclosed within the surface.