Question

Find the potential of a cylindrical conducting can of radius \(a\) and height \(h\) whose top is held at a constant potential \(V_{0}\) while the rest is grounded.

Short answer

The potential of the cylindrical conducting can inside and at the top is \(V_{0}\) while the potential on the surface of the rest of the can is zero.

Step-by-step solution

Understand the given condition

The conducting can is cylindrical with radius \(a\) and height \(h\). Its top is at a constant potential \(V_{0}\) while the rest of the can is grounded, which means its potential is zero. It's important to note that the electric field inside a conductor in static equilibrium is zero.

Apply known principle

This means that the potential is constant throughout the conductor. So, the potential within the cylindrical can is the same as the potential on the top, since there are no other charges within the can whose potential could affect the inside.

Conclude the result

So, the potential of the can surface (with the exception of the top) is zero, since it's grounded. The potential at the top and inside the can is constant and equal to \(V_{0}\) as per the given condition.

Key concepts

Conductors in Electrostatic Equilibrium

Conductors in electrostatic equilibrium are fascinating objects. When in such a state, conductors have unique properties that help us understand electric fields and potentials. One key property is that the electric field inside a conductor is zero when in static equilibrium. This is because free electrons within the conductor redistribute themselves to cancel any internal electric field. For conductors with surfaces at different potentials, such as the cylindrical can in our exercise, even though the top is at potential \( V_0 \), the grounded sides ensure the rest of the conductor maintains potential equilibrium. This distribution allows the interior of the can to be at a uniform potential, with no electric field disturbing the peace inside. This principle is crucial for electrically sensitive environments, where stable currents and potential might need to be maintained.

Electric Potential

Electric potential is a measure of the potential energy per unit charge within an electric field. It is akin to altitude in a gravitational field, where the height represents potential energy. When considering conductors like our cylindrical can, the potential explains the landslide of charges toward equilibrium.In our problem, the top of the cylindrical can is kept at a constant potential \( V_0 \). Since the rest of the can is grounded, it stays at a potential of zero. The entirety of the conductor reaches an electric potential uniformity which means that the potential \( V_0 \) on the can's top surface is the same throughout the inside. This behavior of maintaining a constant potential protects sensitive apparatus inside conductive materials by providing consistent shielding from external electric influences.

Grounding in Electricity

Grounding is an essential concept in electrical circuits and systems. It involves connecting electrical devices or conductors to the Earth to stabilize the voltage. By grounding, you ensure that any excess charge can safely be directed to the ground, thereby balancing the electrical system. In our exercise, grounding comes into play with the grounded sides of the cylindrical can. Grounding these sides effectively sets their potential to zero. This means any excess charge trajectory toward these surfaces flows into the Earth, ensuring no charge builds up on these parts. Grounding is not just a safety measure to protect against electrical overload or damage. It also serves to maintain the desired potential difference, as seen with the can's top being held constant while the remaining body drains into the ground. Understanding grounding helps us design circuits and electrostatic environments that are both safe and predictable.