Problem 31 - University Physics with Modern Physics

Question

22.31. A negative charge \(-Q\) is placed inside the cavity of a hol- low metal solid. The outside of the solid is grounded by connecting a conducting wire between it and the earth. (a) Is there any excess charge induced on the inner surface of the piece of metal? If so, find its sign and magnitude. (b) Is there any excess charge on the outside of the piece of metal? Why or why not?(c) Is there an electric field in the cavity? Explain. (d) Is there an electric field within the metal? Why or why not? Is there an electric field outside the piece of metal? Explain why or why not. (e) Would someone outside the solid measure an electric field due to the charge \(-Q ?\) Is it reasonable to say that the grounded conductor has shielded the region from the ciffects of the charge \(-Q ?\) In principle, could the same thing be done for gravity? Why or why not?

Step-by-Step Solution

Answer

(a) Yes, "+Q" is induced on the inner. (b) No charge on outer, it's grounded. (c) Yes, in cavity. (d) No field in metal; No field outside. (e) No field outside, shielding by conductor; impossible for gravity.

Step 1: Analyze the Situation

We have a negative charge \(-Q\) placed inside the cavity of a hollow metallic solid, with the solid being grounded on the outside. This setup requires us to use principles of electrostatics to determine the behavior of charges and electric fields.

Step 2: Induced Charge on Inner Surface

Since the inner surface of the metal is conducting, a negative charge \(-Q\) inside the cavity induces a positive charge \(+Q\) on the inner surface. This is necessary to maintain electrostatic equilibrium, and ensures that the electric field inside the conductor remains zero.

Step 3: Charge on Outer Surface

Due to grounding, the outer surface of the metal can transfer charge to or from the Earth. The positive charge on the inner surface is balanced by an equal negative charge on the outer surface to maintain the overall neutrality of the metal. Thus, the outer surface does not have any excess charge.

Step 4: Electric Field Inside the Cavity

Inside the cavity, excluding the surface of the cavity itself, there is an electric field present due to the charge \(-Q\) since it creates an electric field pointing towards it.

Step 5: Electric Field Within the Metal

Within the metal, the electric field is zero. This is because, at electrostatic equilibrium, the conductor's interior has no electric field, as any presence of charges would cause movement until the field is neutralized.

Step 6: Electric Field Outside the Metal

Since the outer surface is grounded and bears no excess charge, there is no electric field present outside the metal. Grounding prevents any external electric field from appearing.

Step 7: Measuring Electric Field from Outside and Shielding

There would be no electric field measured from outside due to the charge \(-Q\) because the grounding effectively shields the charge's presence. This grounding effect is specific to electromagnetic forces. However, this shielding cannot occur with gravitational forces, as these cannot be canceled or neutralized by grounding or similar methods.

Key Concepts

Induced Charge

When a charge is introduced into a cavity within a conductor, it causes a phenomenon known as induced charge. In this case, when a negative charge \(-Q\) is placed inside the cavity of a hollow metallic solid, the inner surface of the conductor responds by developing a positive induced charge. This induced charge is of equal magnitude but opposite in sign to the charge \(-Q\). The purpose of this induced charge is to maintain a state of electrostatic equilibrium. By placing a negative charge inside, the negative electric field lines seek equilibrium, thus pulling the positive charges in the conductor towards the inner surface.

This results in a positive charge \(+Q\) being present on the inner surface. Such a response ensures that the potential inside the conductor remains uniform, thereby preventing any electric field inside the metal itself.

Key points about induced charge include:

• The induced charge ensures that the electric field within the conductor's surface remains zero.
• Induced charges depend directly on the charge placed inside the cavity.

Electric Field

The electric field is a vector field that surrounds charged objects and represents the force exerted per unit charge. In electrostatic scenarios, like the one with the hollow conductor, the electric field behaves uniquely both inside the cavity and within the conductor itself.

Inside the cavity, excluding the inner surface, there is an electric field due to the presence of the negative charge \(-Q\). The electric field lines originate from positive charges and move towards negative charges, meaning the field exists surrounding the negative charge inside the cavity.

However, when inside the metal of the conductor, the electric field is zero. This occurs because the charges within the conductor will rearrange themselves to cancel any internal field, achieving electrostatic equilibrium.

Outside the grounded metal, no electric field is measurable because grounding allows any excess charge to dissipate. As a result, external observers detect no electric field emanating from the hollow conductor.

These properties highlight the behavior of electric fields in and around conductors and underscore the role of grounding in managing electric fields within electrical environments.

Grounded Conductor

Grounding is the process of connecting a conductor to the Earth, enabling it to receive or dissipate charge as needed. When the outer surface of the hollow conductor in question is grounded, it plays a critical role in maintaining the conductor's neutrality.

By grounding the conductor, excess charge on the outer surface can be transferred to or from the Earth. In this scenario, since a positive charge \(+Q\) is induced on the inner surface, a corresponding negative charge \(-Q\) is facilitated through grounding on the outer surface, thereby maintaining the overall charge neutrality of the conductor.

This setup leads to no net charge seen from the outside, and the grounded conductor effectively shields the outer region from the electric field effects of the internal charge \(-Q\). This shielding is invaluable in controlling electrical environments, making grounded conductors essential in many applications. However, it's important to note that while grounding can neutralize electric fields, it does not affect gravitational fields.

Electrostatic Equilibrium

Electrostatic equilibrium is a state where the net electric field inside a conductor is zero and the charge distribution remains static. In the hollow metal conductor scenario, achieving electrostatic equilibrium is crucial, particularly inside the metal.

Due to the presence of a negative charge \(-Q\) inside the cavity, the inner surface induces a positive charge \(+Q\). This redistribution ensures that the electric field inside the conductor's metal remains zero.

At equilibrium:

• The electric field within the conducting material cancels out completely.
• All excess charge resides only on the surface.

Electrostatic equilibrium explains why charges distribute themselves on the surfaces of conductors and why the interior of a conductor can have no electric field. Through grounding, the outflow of charge keeps the outer surface neutral, reinforcing the equilibrium state and ensuring the conductor effectively shields against the internal charge.