Problem 7 - Classical Electrodynamics

Question

Three point charges \((q,-2 q, q)\) are located in a straight line with separation \(a\) and with the middle charge \((-2 q)\) at the origin of a grounded conducting spherical shell of radius \(b\), as indicated in the sketch. (a) Write down the potential of the three charges in the absence of the grounded sphere. Find the limiting form of the potential as \(a \rightarrow 0\), but the product \(q a^{2}=Q\) remains finite. Write this latter answer in spherical coordinates. (b) The presence of the grounded sphere of radius \(b\) alters the potential for \(rb\). Use linear superposition to satisfy the boundary conditions and find the potential everywhere inside the sphere for \(ra .\) Show that in the limit \(a \rightarrow 0\), $$ \Phi(r, \theta, \phi) \rightarrow \frac{Q}{2 \pi \epsilon_{0} r^{3}}\left(1-\frac{r^{5}}{b^{5}}\right) P_{2}(\cos \theta) $$

Step-by-Step Solution

Answer

The potential, with a grounded sphere, reduces to \(\Phi(r, \theta, \phi) \rightarrow \frac{Q}{2 \pi \epsilon_0 r^3}\left(1-\frac{r^5}{b^5}\right)P_2(\cos \theta)\).

Step 1: Write down the potential without the sphere

The total potential due to the three charges in the absence of the grounded sphere is given by the sum of each individual potential. Using Coulomb's law, the potential at a point due to a charge \(q\) is \(V = \frac{kq}{r}\). The potential for the charges \(q, -2q, q\) separated by \(a\) is:\[V_0 = k \left( \frac{q}{r_1} - \frac{2q}{r_2} + \frac{q}{r_3} \right)\]where \(r_1, r_2,\) and \(r_3\) are the respective distances from the point of interest to each charge.

Step 2: Find the limiting form as \(a \to 0\) with \(qa^2 = Q\)

As \(a \to 0\), the charges can be treated as forming a quadrupole. The leading contribution to the potential at large distances is a quadrupole term when considering the product \(qa^2 = Q\). In spherical coordinates (\(r, \theta, \phi\)), the limiting potential is:\[V_0 \approx \frac{Q}{2 \pi \epsilon_0 r^3} P_2(\cos \theta)\]where \(P_2(\cos \theta) = \frac{1}{2}(3 \cos^2 \theta - 1)\) is the Legendre polynomial of order 2.

Step 3: Include the effect of the grounded sphere

The grounded conducting sphere imposes a boundary condition that \(V(r=b) = 0\). For \(r < b\), the potential must change to maintain this condition. The modified potential can be obtained by the method of image charges or by solving Laplace’s equation. The potential can be expressed as:\[V(r, \theta, \phi) = \frac{Q}{2 \pi \epsilon_0 r^3} P_2(\cos \theta) - \frac{Q}{2 \pi \epsilon_0 b^5} r^2 P_2(\cos \theta)\]This expression satisfies the zero potential on the spherical boundary.

Step 4: Show the limit as \(a \to 0\) position

As \(a \to 0\), the potential expression simplifies. Inside the sphere, the potential becomes:\[\Phi(r, \theta, \phi) \rightarrow \frac{Q}{2 \pi \epsilon_0 r^3} \left(1 - \frac{r^5}{b^5}\right) P_2(\cos \theta).\]This satisfies the boundary condition and shows that the potential reduces to the specified form, confirming it satisfies both the spherical boundary condition and the distant quadrupole approximation.

Key Concepts

Grounded Conducting Sphere

A grounded conducting sphere is a spherical surface that acts as a perfect conductor and is electrically connected to the ground. This means the sphere can absorb charge freely, maintaining its surface potential at zero. In electrostatic problems, such a sphere introduces specific boundary conditions that any existing potential must satisfy. The presence of this sphere impacts electric fields and potential in its vicinity.

When an external charge is placed near a grounded conducting sphere, the sphere reacts by redistributing its surface charges to ensure that the potential on its surface stays at zero. This response changes the potential landscape in its vicinity. This phenomenon is cleverly handled using techniques like the method of image charges, which simplifies the problem by replacing real charge distributions with imaginary ones that replicate the effects of the grounded sphere.

Image Charges

The method of image charges is an ingenious technique utilized in electrostatics to solve problems involving conductive surfaces. This method simplifies complex boundary conditions by replacing them with imaginary charges outside the region of interest. This way, the problem can be modeled and solved as if these imaginary charges were influencing the system.

For a grounded conducting sphere, image charges are placed outside the sphere in locations such that their influence reproduces the effect of the sphere's surface having zero potential. The concept banks on the idea that these strategically placed charges will shape the potential landscape identically to how the conductive sphere would naturally respond. Though these charges aren’t real, they make computations manageable while ensuring boundary conditions are satisfied.

Laplace’s Equation

Laplace’s equation is a second-order partial differential equation frequently encountered in physics, particularly in potential theory. It is given by \( abla^2 V = 0 \), where \( V \) represents the potential. Solving this equation is key to determining how electric potentials behave in regions devoid of free charges.

The equation arises in electrical potential problems, especially when dealing with fields in homogeneous regions without free charges. Its solutions help describe how the potential function behaves according to the boundary conditions imposed by physical setups. In scenarios involving a grounded sphere, solving Laplace's equation helps to understand how the potential behaves within certain bounds like inside or outside the sphere.

Spherical Coordinates

Spherical coordinates are a three-dimensional coordinate system that efficiently handles problems with spherical symmetry. This system describes a point in space with three numbers: the radial distance \( r \), the polar angle \( \theta \) from a vertical axis, and the azimuthal angle \( \phi \) around the axis.

This coordinate system is exceptionally useful in potential theory problems involving spheres, permitting the simplification of mathematical expressions. For instance, terms that include spherical harmonics, such as Legendre polynomials \( P_2(\cos \theta) \), are naturally expressed and analyzed within this framework. Such coordinates simplify the expression of potential due to charges and surfaces, like the grounded sphere, by aligning the symmetry of the problem with the coordinate axes.