Question

Suppose that the Faraday concentric sphere experiment is performed in free space using a central charge at the origin, \(Q_{1}\), and with hemispheres of radius a. A second charge \(Q_{2}\) (this time a point charge) is located at distance \(R\) from \(Q_{1}\), where \(R>>a .(a)\) What is the force on the point charge before the hemispheres are assembled around \(Q_{1} ?\) (b) What is the force on the point charge after the hemispheres are assembled but before they are discharged? ( \(c\) ) What is the force on the point charge after the hemispheres are assembled and after they are discharged? ( \(d\) ) Qualitatively, describe what happens as \(Q_{2}\) is moved toward the sphere assembly to the extent that the condition \(R>>a\) is no longer valid.

Short answer

Answer: The force on Q2 remains constant and can be calculated using Coulomb's force formula (F = k(Q1Q2)/R^2) in the scenarios before the assembly, after the assembly, and after the discharging of the hemispheres. However, when Q2 moves closer to the sphere assembly, the net force on Q2 changes due to the influence of the charged hemispheres and its position relative to the assembly.

Step-by-step solution

The force on the point charge Q2 can be calculated using Coulomb's force formula: \(F = k\frac{Q_1Q_2}{R^2}\). Since the conditions given in the problem are \(Q_1\), \(Q_2\), and \(R\), we can use the given values to compute the force on Q2 before the hemispheres are assembled. #b) After the hemispheres are assembled but before they are discharged:

Since the hemispheres do not have charges on them before they are discharged, and they are assembled symmetrically around Q1, the force on Q2 remains unchanged. This is because the electric field created by Q1 on Q2 would not be altered by the presence of the hemispheres. Hence, the force on Q2 in this case is the same as in part (a): \(F = k\frac{Q_1Q_2}{R^2}\). #c) After the hemispheres are discharged:

When the hemispheres are discharged, they obtain charges. However, since they are symmetrically placed around Q1 and at a greater distance than Q2, the electric field produced by these charges would cancel each other out. Thus, even after discharging the hemispheres, the force on Q2 remains the same: \(F = k\frac{Q_1Q_2}{R^2}\). #d) Describing what happens as Q2 moves closer to the sphere assembly:

When Q2 moves closer to the sphere assembly, and the condition \(R >> a\) is no longer valid, the impact of the charged hemispheres on Q2 becomes significant. As Q2 approaches the sphere assembly, the force between Q2 and each charged hemisphere would not cancel each other out anymore. This will result in a different net force acting on Q2. Additionally, the shape of the electric field lines near the sphere assembly will change due to their influence, causing a change in the force experienced by Q2. Overall, the force on Q2 will depend on its position relative to the sphere assembly and the charges on the hemispheres.

Key concepts

Coulomb's force

Coulomb's force is a fundamental principle in electromagnetics that describes the interaction between charged particles. It is used to calculate the force between two point charges.

• The formula to determine the force is: \( F = k \frac{Q_1 Q_2}{R^2} \).
• Here, \( F \) represents the force between the charges, \( k \) is Coulomb's constant, \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges, and \( R \) is the distance between the charges.

Understanding this force is crucial for determining how particles interact at a distance. The force can be attractive if the charges are opposite, or repulsive if the charges are similar.In the exercise, Coulomb's force is used to calculate the initial force on the point charge \( Q_2 \) before any assemblies or discharges occur. This fundamental principle helps in understanding the interactions and forces at play in systems involving charged particles.

Faraday's experiment

Faraday's experiment with concentric spheres provides insights into how electric fields behave in unique scenarios. His work laid the groundwork for understanding how charges distribute themselves under specific conditions.

• In the case of the hemispheres surrounding a central charge \( Q_1 \), they create a unique distribution of electric fields.
• The experiment shows that before the hemispheres are discharged, their presence does not alter the force on a point charge \( Q_2 \), as the electric field created by \( Q_1 \) remains undisturbed.
• After discharging, any charges the hemispheres may have cancel each other out due to the symmetric arrangement.

Faraday's insights are particularly useful in understanding potential shielding effects and the behavior of fields within complex geometries.

Electric field

An electric field is a region around a charged particle where other charged particles experience a force. It is a vector field, meaning it has both magnitude and direction.

• The strength and direction of the electric field are determined by the source charge and distance from it.
• Mathematically, the electric field \( E \) can be defined as: \( E = \frac{F}{Q} \), where \( F \) is the force experienced by a test charge \( Q \).
• In the context of the exercise, the electric field produced by the central charge \( Q_1 \) influences the point charge \( Q_2 \).

When hemispheres are introduced, they can alter the distribution and effects of the electric field. However, due to their symmetry, the net effect on \( Q_2 \) before discharging remains the same, illustrating the field's dependence on spatial arrangements and shapes.

Point charge interaction

Point charge interaction refers to how individual charged particles affect each other within an electric field.

• Interactions are driven by the forces calculated using Coulomb's law, which depend on the magnitudes of the charges and their separation distance.
• Before the hemispheres are assembled, \( Q_2 \) experiences a force due to the electric field from \( Q_1 \).
• Once the hemispheres are assembled and discharged, the interaction involves additional considerations, such as the distribution of the field from the charged hemispheres.

This interaction becomes more complex when the point charge \( Q_2 \) moves closer to the sphere assembly, as the resulting field forces and configurations shift. Understanding these interactions is crucial for predicting behavior in electromagnetic environments and fields.