Question

Construct a curvilinear-square map of the potential field between two parallel circular cylinders, one of \(4 \mathrm{~cm}\) radius inside another of \(8 \mathrm{~cm}\) radius. The two axes are displaced by \(2.5 \mathrm{~cm}\). These dimensions are suitable for the drawing. As a check on the accuracy, compute the capacitance per meter from the sketch and from the exact expression: $$C=\frac{2 \pi \epsilon}{\cosh ^{-1}\left[\left(a^{2}+b^{2}-D^{2}\right) /(2 a b)\right]}$$ where \(a\) and \(b\) are the conductor radii and \(D\) is the axis separation.

Short answer

Answer: The relationship between the capacitance and the number of squares is given by the formula: $$C=\frac{\epsilon N}{d}$$ where \(C\) is the capacitance per meter, \(\epsilon\) is the permittivity of the medium, \(N\) is the number of squares between the conductors, and \(d\) is the distance between the conductors.

Step-by-step solution

Determine the potentials on the cylinders

Given the problem, consider two parallel circular cylinders: one with a radius of \(4\mathrm{~cm}\) and the other with a radius of \(8\mathrm{~cm}\). Their axes are displaced by \(D=2.5\mathrm{~cm}\). Let's take the inner conductor as our reference and assign a potential of \(V=0\) to the inner conductor. We can assign a potential of \(V=1\) to the outer conductor.

Draw a curvilinear-square map of the potential field

To draw the curvilinear-square map, we need to find the equipotential lines, which are the paths of constant potential. The potential lines can be drawn perpendicular to the equipotential lines. Then we can visually create squares using these lines. - Start by drawing the two cylinders with the given dimensions. - Draw equipotential lines between the cylinders representing various potential values (e.g., \(0.2, 0.4, 0.6, 0.8\)). - Draw potential lines perpendicular to the equipotential lines, thus creating squares. Keep in mind, it's vital to make the sketch as accurate as possible since we will be using it to find the capacitance per meter.

Find the capacitance per meter using the sketch

To estimate the capacitance using the curvilinear-square map, we need to count the number of squares between the inner and outer conductors and use the relation between capacitance and the number of squares: $$C=\frac{\epsilon N}{d}$$ where \(C\) is the capacitance per meter, \(\epsilon\) is the permittivity of the medium, \(N\) is the number of squares between the conductors, and \(d\) is the distance between the conductors. Count the number of squares between the inner and outer conductors and use this value for \(N\).

Compute the capacitance per meter using the exact expression and compare the results

Next, we use the exact expression provided: $$C=\frac{2 \pi \epsilon}{\cosh ^{-1}\left[\left(a^{2}+b^{2}-D^{2}\right) /(2 a b)\right]}$$ Inserting the given values for \(a=4\mathrm{~cm}\), \(b=8\mathrm{~cm}\), and \(D=2.5\mathrm{~cm}\): $$C=\frac{2 \pi \epsilon}{\cosh ^{-1}\left[\left(4^{2}+8^{2}-2.5^{2}\right) /(2 (4)(8))\right]}$$ Calculate the capacitance \(C\) using the formula above. Once you have the capacitance calculated using both methods, compare the results to check the accuracy of the sketch. Since the sketch is an approximation, both the values won't exactly match, but it should be reasonably close if the sketch is accurate.

Key concepts

Curvilinear-square map

To understand a curvilinear-square map, imagine superimposing a square grid onto a curved surface. This technique is used in the field of electrostatics to visually represent fields between two conductors with non-linear shapes, like cylinders. In this case, you create a map to depict potential fields between two parallel circular cylinders.

The idea is to create a grid where potential lines are perpendicular to equipotential lines, forming approximate "squares" in the region of interest. By extending these principles, you can convert complicated spatial arrangements into simpler geometrical interpretations. This method plays a critical role in visualizing potential distributions, thus aiding calculations related to parameters like capacitance.

Equipotential lines

Equipotential lines are an essential concept when dealing with fields between conductors. These lines are the loci of points with the same electric potential. In simpler terms, if you move along an equipotential line, you won’t experience any voltage difference.

In our example with the cylinders, the equipotential lines curve between the two cylinders. They are essential to constructing the curvilinear-square map because they provide a reference for plotting potential lines that are perpendicular and help in visualizing the electric field 'flow.'

It's important to understand that the density of these lines indicates the field strength: the closer they are, the stronger the electric field.

Capacitance per meter

Capacitance per meter is a measure of a system's ability to store charge per unit length. It is particularly useful when evaluating long cylindrical conductors like cables.

This concept can be determined visually using the curvilinear-square map by counting the number of squares between the inner and outer conductors. The formula involved, \[ C = \frac{\epsilon N}{d} \]relates capacitance per unit length, permittivity \( \epsilon \), the number of squares \( N \), and the distance \( d \) between conductors.

In practical terms, having a high capacitance per meter suggests that the conductor can store a large amount of electric charge, which is a key quality in many applications involving electrical transmission and storage.

Permittivity

Permittivity is a property of materials that affects how electric fields interact with them. It signifies how much electric field energy can be stored in a material. In equations, it often appears as \( \epsilon \).

In our cylindrical conductor example, permittivity determines the capacitance between the cylinders. Materials with higher permittivity lead to a higher capacitance. This makes selecting the right materials crucial in designing systems that require specific electrical storage capabilities.

Permittivity also dictates how external fields affect the material, making it fundamental in electrical engineering and material science.

Exact expression for capacitance

The exact expression for capacitance provides a precise calculation, considering all defined parameters of the system. For the geometry of parallel circular cylinders, it’s given by the formula:\[ C = \frac{2 \pi \epsilon}{\cosh^{-1}\left(\frac{a^2 + b^2 - D^2}{2ab}\right)} \]where \( a \) and \( b \) are the radii of the cylinders and \( D \) is the distance between their centers.

Using this formula, one can compute the theoretical capacitance and compare it with estimates obtained from visual methods like the curvilinear-square map to assess accuracy. The closer these two values are, the more accurate your visual representation and assumptions were.

This concept underscores the necessity for precise calculations alongside visual approximations in engineering disciplines.