Question

A sphere of radius \(a\) carries a surface charge density of \(\rho_{s 0} \mathrm{C} / \mathrm{m}^{2} .(a)\) Find the absolute potential at the sphere surface. ( \(b\) ) A grounded conducting shell of radius \(b\) where \(b>a\) is now positioned around the charged sphere. What is the potential at the inner sphere surface in this case?

Short answer

If the sphere is surrounded by a grounded conducting shell with radius \(b\), what is the potential at the inner sphere surface? Answer: The potential at the surface of the charged sphere is given by \(V = \frac{k\rho_{s0}(4\pi a^2)}{a}\), where \(k\) is the electrostatic force constant. When the grounded conducting shell with radius \(b\) is placed around the sphere, the potential at the inner sphere surface becomes \(V = \frac{kQ}{b}\).

Step-by-step solution

Finding the charge on the sphere

First, we'll find the total charge on the sphere. The surface charge density is \(\rho_{s0} \mathrm{C} / \mathrm{m}^{2}\). We can find the total charge by multiplying the surface charge density by the area of the sphere: \(Q = \rho_{s0} \cdot A\) The area of a sphere is given by \(A = 4\pi a^2\), where \(a\) is the radius. So, we have \(Q= \rho_{s0}(4\pi a^2)\)

Finding the absolute potential at the sphere surface (for part a)

In electrostatics, the potential \(V\) at a distance \(r\) from a point charge with charge \(Q\) is given by: \(V = \frac{kQ}{r}\) where \(k= \frac{1}{4\pi \epsilon_0}\) is the electrostatic force constant and \(\epsilon_0\) is the permittivity of free space. In our case, the distance \(r\) is equal to the radius of the sphere \(a\). So we can find the potential at the sphere surface as: \(V = \frac{k \rho_{s0}(4\pi a^2)}{a}\)

Considering the grounded conducting shell

Now we need to find the potential at the inner sphere surface when it's surrounded by a conducting shell of radius \(b\) with \(b>a\). The conducting shell is grounded, which means its potential is 0 V. Since the conducting shell is grounded, it will induce an opposite charge on its inner surface. This induced charge can be represented as an image charge. We can apply the principle of superposition of potentials to find the potential at the inner sphere surface.

Finding the potential at the inner sphere surface (for part b)

Since the outer shell is a conductor, the total potential inside the conductor is zero. Considering the potential due to the charge \(Q\) on the inner sphere and the potential due to the image charge \(Q_{image}\) on the outer shell, we have: \(V_{image} = \frac{kQ_{image}}{a} = \frac{kQ}{b}\) Now, we can solve for \(Q_{image}\): \(Q_{image} = \frac{a}{b}Q\) Thus the potential at the inner sphere surface when surrounded by the grounded conducting shell is given by: \(V = \frac{kQ_{image}}{a} = \frac{k\frac{a}{b}Q}{a} = \frac{kQ}{b}\)

Key concepts

Surface Charge Density

Surface charge density is a fundamental concept in electrostatics. It measures the amount of electric charge per unit area on a surface and is denoted by symbols like \( \rho_{s} \). Imagine spreading a thin layer of charge over a balloon's surface; the amount of charge per square meter is what we call the surface charge density.

When solving electrostatic problems, understanding surface charge density allows us to calculate the total charge distributed over an object's surface. Here, for a sphere with radius \(a\), the charge \(Q\) on the sphere's surface would be the product of the surface charge density \( \rho_{s0} \) and the sphere's surface area \(A\). Given that the area of the sphere is \(4\pi a^2\), we calculate the total charge as \(Q= \rho_{s0}(4\pi a^2)\).

Knowing the charge and the geometry allows us to explore electric fields and potential created by this distribution, essential for understanding the behavior of charges and fields in electrostatic situations.

Grounded Conducting Shell

A grounded conducting shell is an important concept that frequently arises in electrostatics and electrical engineering. When we say a conductor is grounded, we mean it is physically connected to the Earth or another large body that can act as an infinite reservoir of charge. Grounding ensures that the potential of the conductor is equal to zero.

What happens when you place a charged object near or inside a grounded conducting shell? The shell responds by redistributing its free charges such that it cancels out any electric field within its thickness. This results in an induced charge on the inner surface of the shell that exactly matches and neutralizes the effect of the internal charge's field.

Therefore, if you have a sphere with charge inside a grounded conducting shell, the potential within the shell and on its surface would be zero, a crucial point when solving problems regarding electrostatic potentials in the presence of conductors.

Image Charge Method

The image charge method is a powerful technique used in electrostatics to solve complex problems involving conductors. It employs the concept of imaginary charges (image charges) that, when placed at strategic positions, replicate the effects of conductive boundaries like a conducting shell.

This method is particularly useful for grounded conductors. For instance, a point charge near a grounded conducting plane can be analyzed by imagining an equal and opposite charge located symmetrically on the other side of the plane. This 'image charge' generates a field that, when combined with the actual charge, adheres to the boundary conditions imposed by the grounded conductor—that the potential at the surface of the conductor must be zero.

By using the image charge method, complex problems are reduced to simpler ones involving only point charges, significantly simplifying the computation of electric fields and potentials.

Electric Field and Potential

The concepts of electric field and potential are crucial to understanding how electric forces behave and interact in space. The electric field, denoted by \( \vec{E} \), is a vector quantity representing the force experienced by a unit positive charge at a point in space.

The electric potential, on the other hand, is a scalar quantity that measures the electric potential energy per unit charge at a point, with the symbol \( V \). It informs us about the work done in bringing a charge from infinity to that point without any acceleration.

In our example, we calculate the electric potential at the surface of a charged sphere and subsequently determine how it changes when this sphere is encased within a grounded conducting shell. The potential \(V\) due to a point charge is given by \(V = \frac{kQ}{r}\), with \(k = \frac{1}{4\pi \epsilon_0}\), where \(Q\) is the charge, \(r\) is the distance from the charge, and \(\epsilon_0\) is the permittivity of free space. For spherical conductors and shells, it's important to remember that these concepts help predict how electric charges will interact with their surroundings.