Question

A small conducting sphere of radius \(r\) is lying concentrically inside a bigger hollow conducting sphere of radius \(R\). The bigger and smaller sphere are charged with \(\mathrm{Q}\) and \(\mathrm{q}(\mathrm{Q}>\mathrm{q})\) and are insulated from each other. The potential difference between the spheres will be \(\ldots \ldots\) (A) \(\left[1 /\left(4 \pi \epsilon_{0}\right)\right][(q / r)-(Q / R)]\) (B) \(\left[1 /\left(4 \pi \epsilon_{0}\right)\right][(q / r)-(q / R)]\) (C) \(\left[1 /\left(4 \pi \epsilon_{0}\right)\right][(Q / R)+(q / r)]\) (D) \(\left[1 /\left(4 \pi \epsilon_{0}\right)\right][(\mathrm{q} / \mathrm{R})-(\mathrm{Q} / \mathrm{r})]\)

Short answer

The potential difference between the spheres is: \[V = \left[\frac{1}{4 \pi \epsilon_0}\right] \left[\frac{Q}{R}-\frac{q}{r}\right]\]

Step-by-step solution

Find potential at the surface of the inner sphere

To find the potential at the surface of the inner sphere, we use the formula V = kQ/r, where Q = q (charge on the inner sphere) and r is its radius. V_inner = k * (q / r)

Find potential at the surface of the outer sphere

Similarly, we will now find the potential at the surface of the outer sphere V_outer using its charge (Q) and radius (R). V_outer = k * (Q / R)

Calculate the potential difference

Now, we will find the potential difference between the two spheres (V_inner and V_outer) by subtracting them. Remember, the potential difference is given by V = V_outer - V_inner. V = V_outer - V_inner V = k * (Q / R) - k * (q / r) Now, factor out k: V = k * [(Q / R) - (q / r)] We know that k equals 1 / (4 * pi * ε₀), so we can substitute that: V = (1 / (4 * pi * ε₀)) * [(Q / R) - (q / r)] Comparing this with the given options, we can see that the potential difference between the spheres is equal to option (A).

Key concepts

Potential Difference

The potential difference is a measure of the work needed to move a unit electric charge from one point to another within an electric field. In simple terms, it explains how much energy is needed to move a charge between two points.
A higher potential difference indicates that more work is required, and thus the points have a greater difference in electric potential energy. This can be likened to the flow of water; water flows naturally from a higher to a lower elevation, just like electric charges move due to potential differences in a circuit or system.

• It's often measured in volts (V).
• This concept is crucial in electrical circuits where devices operate based on potential differences across their terminals.
• Here, between the two spheres, it is calculated using the difference in potential energy per unit charge on each sphere's surface.

Conducting Sphere

A conducting sphere, in electrostatics, serves as a classic model to understand how charges distribute over a surface. Conductors have materials that allow free flow of electric charges across their surface and interior. In the case of a spherical conductor, the charges reside entirely on the surface due to repulsion among like charges.
When dealing with two concentric spheres, like in this exercise, each sphere can have charges that do not affect the charge distribution on the interior surface, thanks to the shielding property of conductors.

• Charges are evenly distributed on the outer surface of the sphere.
• Spheres in electrostatic equilibrium ensure no net motion of charge inside the conductor.
• In this problem, the potential at different locations on or around the conducting spheres results from these surface charges.

Coulomb's Law

Coulomb's Law is fundamental in understanding electric forces between charged objects. It defines how the electric force between two point charges is:

• Directly proportional to the product of the magnitudes of charges.
• Inversely proportional to the square of the distance between their centers.

Using the constant of proportionality and substituting values, you can predict the magnitude of electrostatic forces in various configurations. In this exercise, Coulomb's Law helps in calculating potential energies as these relate to distances, radii, and magnitudes of charges. Interestingly, even though this law speaks primarily to forces, its concepts translate into understanding charges and potentials, directly impacting how we calculate potential differences between spheres.

Electric Potential

Electric potential is like the electric version of gravitational potential energy. Imagine it as the amount of potential energy per unit charge at a point in space due to electric forces. In simpler terms, it's the capability of a field to do work on a charge, and it's often measured in volts (V).
The electric potential is central to this problem, as it determines the potential difference, which we evaluate between two charged spheres.

• Potential on a sphere is calculated using the charge and the radius of the sphere because potential energy translates from quantities related to electricity.
• For example, for each sphere, the formula used is V = kQ/r, where "k" is the constant \(1/(4 \pi \varepsilon_0)\).
• Consequently, knowing individual potentials helps find the potential difference, which is straightforward as subtracting one from the other.

Understanding this concept ensures that you comprehend how electric fields store energy and affect charges within a field.