Question

The lateral faces of a cube are grounded, and its top and bottom faces are held at potentials \(f_{1}(x, y)\) and \(f_{2}(x, y)\), respectively. (a) Find a general expression for the potential inside the cube. (b) Find the potential if the top is held at \(V_{0}\) volts and the bottom at \(-V_{0}\) volts.

Short answer

The general expression for the potential inside the cube is \(V(x, y, z) = \frac{z}{a}f_{2}(x, y) + (1 - \frac{z}{a}) f_{1}(x, y)\). If the top is held at \(V_{0}\) volts and the bottom at \(-V_{0}\) volts, the potential across the cube is \(V(x, y, z) = 2V_{0}\left (\frac{z}{a}-\frac{1}{2} \right)\).

Step-by-step solution

- General Expression for Potential

Let’s refer the potentials at the cube's top and bottom faces as \(f_{1}(x, y)\) and \(f_{2}(x, y)\) respectively and denote the cube’s side length by \(a\). Let \(z\) be the height above the ground. We can obtain a general expression for the potential \(V(x, y, z)\) inside the cube by solving the Laplace's equation \(\nabla ^{2}V = 0\), with boundary conditions of \(V(x, y, 0) = f_{1}(x, y)\) and \(V(x, y, a) = f_{2}(x, y)\). The solution will vary depending on the specifics of the functions \(f_{1}(x, y)\) and \(f_{2}(x, y)\). However, a general form can be written as: \\\( V(x, y, z) = \frac{z}{a}f_{2}(x, y) + (1 - \frac{z}{a}) f_{1}(x, y) \\

- Specific Potential Value

Next step is to find a more specific expression for the potential when the cube's faces are held at \(V_{0}\) and \(-V_{0}\) respectively. By substituting these values into the general expression for potential, ie; \(f_{1}(x, y) = -V_{0}\) and \(f_{2}(x, y) = V_{0}\), we get: \\ \(V(x, y, z) = \frac{z}{a}V_{0} - V_{0} + V_{0}\\V(x, y, z) = 2V_{0}\left (\frac{z}{a}-\frac{1}{2} \right) \\

- Conclusion

The potential inside the cube will vary depending upon the functions defining the top and bottom potentials. These functions also give way to an equation that represents the electric field within the cube. Even in a specific case, the potential will depend upon the height from the bottom face. The given set of values provides an understanding about how potential changes according to different points within the cube.

Key concepts

Laplace's equation

Imagine you're trying to find the electrostatic potential at every point inside a cube, a space where there are no charges present. This might seem daunting, but physicists and mathematicians have a powerful tool for this situation: Laplace's equation. This mathematical formula, written as abla^2 V = 0, is a way of stating that in regions devoid of charges, the potential doesn't have any sudden jumps or spikes. It essentially tells us how potential is smoothly distributed in a space.

Laplace's equation is a type of partial differential equation (PDE), which describes how a quantity like electrostatic potential changes in space. Solving this equation isn't always straightforward since the solutions vary with the shape of the space and the potential values on the boundaries. But the good news is, if you know the potential values along the boundaries of the space—like the faces of a cube solution is completely possible.

Boundary conditions

The concept of boundary conditions is a little like setting up rules for a game before you start playing. When solving for potential inside a cube using Laplace's equation, boundary conditions tell you the potential values on the surfaces of the cube. They're essential because they define the electrostatic playground upon which potential can dance.

In our cube example, the lateral faces are grounded, which means they're set to zero volts; this is one type of boundary condition. The top and bottom faces, however, are held at arbitrary functions f₁(x, y) and f₂(x, y). These prescribed conditions guide the mathematics to a unique solution for the potential within the cube. Without boundary conditions, you could come up with an infinite number of potential distributions—all mathematically correct, but not all physically meaningful.

Electrostatics

Electrostatics is a branch of physics that deals with the study of stationary or slow-moving electric charges. It's the science beneath sparks when you touch a doorknob or why your hair might stand up when you pull a wool hat off your head. When working with an exercise involving a cube with particular electrostatic potentials on its faces, you're dipping your toes into the calm waters of electrostatics.

In electrostatics, electric fields and potentials describe the electric force landscape. A key takeaway is that electric fields originate from electric charges, but in an electrostatic scenario within a conductor, like the cube's grounded sides, the field is zero, leading to constant potential, often taken as zero. Grounded surfaces in an electrostatic problem serve as references, simplifying the search for the mysterious potential at each point inside your cube.

Mathematical physics

Mathematical physics intertwines mathematics and physics, using mathematical methods to solve physical problems. It's a bridge allowing us to cross from abstract math equations to concrete physical phenomena. When working on the cube's potential, we stride across this bridge.

Understanding the potential inside a cube involves applying concepts from mathematical physics, such as solving differential equations under certain conditions (those boundary conditions we talked about). These techniques are not just useful academically; they have real-world applications in fields ranging from engineering to meteorology, anywhere the invisible forces of nature need to be understood or harnessed.

Potential function

A potential function is like a map that tells you the electric potential landscape of a region—in other words, how much electric potential energy a test charge would have at every point in space. When a potential function is known, you can often easily figure out the electric field by taking the negative gradient of the potential.

In this cube exercise, the potential function describes how the electric potential changes from one point to another inside the cube. For points between the top and bottom faces with potentials f₁(x, y) and f₂(x, y), the potential function guides us mathematically from one 'height' of potential to another, just like an algebraic elevator. By understanding the potential function, we learn not just the intensity of potential at each point, but also how this potential changes and interacts with the electric field.