Question

The voltage across a cylindrical conductor of radius \(r\), length \(L\), and resistance \(R\) varies with time. The timevarying voltage causes a time-varying current, \(i\), to flow in the cylinder. Show that the displacement current equals \(\epsilon_{0} \rho d i / d t,\) where \(\rho\) is the resistivity of the conductor.

Short answer

Question: Prove that the displacement current in a cylindrical conductor equals \(\epsilon_0 \rho \frac{di}{dt}\), where \(\rho\) is the resistivity of the conductor. Answer: The proof involves using the relationship between resistance, resistivity, and the geometry of the conductor, as well as Ohm's Law. By finding the resistance of the cylindrical conductor, determining the current, and then finding the time derivative of the current, we could show that the expression for displacement current, \(i_D = \epsilon_0 \frac{1}{\rho \frac{L}{\pi r^2}} \frac{dV}{dt}\), simplifies to the given expression, \(\epsilon_0 \rho \frac{di}{dt}\).

Step-by-step solution

Determine the Resistance

The resistance of a cylindrical conductor can be determined using the formula: $$R = \rho \frac{L}{A},$$ where \(A\) is the cross-sectional area of the conductor, \(L\) is its length, and \(\rho\) is its resistivity. The cross-sectional area of a cylinder is simply the area of a circle, which is: $$A = \pi r^2,$$ where \(r\) is the radius of the cylinder. Plugging this into the resistance formula, we get: $$R = \rho \frac{L}{\pi r^2}.$$

Determine the Current

Ohm's Law states that the voltage across a conductor is related to the current flowing through it and its resistance: $$V = I R,$$ where \(V\) is the voltage across the conductor, \(I\) is the current through the conductor, and \(R\) is its resistance. Thus, we can find the current flowing through the conductor as follows: $$I = \frac{V}{R}.$$ Using the formula for the conductor's resistance from Step 1, we get: $$I = \frac{V}{\rho \frac{L}{\pi r^2}}.$$

Determine the Displacement Current

The displacement current is given by the expression: $$i_D = \epsilon_0 \frac{dI}{dt},$$ where \(\epsilon_0\) is the vacuum permittivity constant and \(\frac{dI}{dt}\) is the time derivative of the current. To find \(\frac{dI}{dt}\), we differentiate the current expression from Step 2 with respect to time: $$\frac{dI}{dt} = \frac{1}{\rho \frac{L}{\pi r^2}} \frac{dV}{dt}.$$

Show that the Displacement Current Equals the Given Expression

Now, we will plug the expression for \(\frac{dI}{dt}\) from Step 3 into the displacement current formula and see if it matches the given expression: $$i_D = \epsilon_0 \frac{1}{\rho \frac{L}{\pi r^2}} \frac{dV}{dt} = \epsilon_0 \rho \frac{di}{dt}.$$ And this is the given expression for displacement current that we were asked to prove.

Key concepts

Resistivity

When working with electrical circuits, it's essential to understand resistivity, which is a fundamental property of materials. Resistivity, often symbolized as \( \rho \), measures how strongly a material opposes the flow of electric current. It's determined by the type of material and is influenced by temperature. In a practical scenario, the resistivity of a conductor affects how much resistance it has for a given size.

For a cylindrical conductor, as seen in the original exercise, the resistance \( R \) can be calculated using the resistivity \( \rho \) along with the conductor's length \( L \) and its cross-sectional area \( A \) through the formula \( R = \rho \frac{L}{A} \). Understanding \( \rho \) is critical, as it directly impacts how we develop electrical components and choose materials for conducting electrical current.

Ohm's Law

Ohm's Law is one of the most important principles within electrical engineering and physics. It explains the relationship between voltage \( V \), current \( I \), and resistance \( R \) in an electrical circuit: \( V = IR \). This means that the voltage across a conductor is directly proportional to the current flowing through it, and inversely proportional to its resistance.

This relationship enables us to calculate any one of these three variables if the other two are known. In the case of a cylindrical conductor with a time-varying voltage, Ohm's Law helps in determining the current \( I \) that results from a given voltage \( V \) applied across the conductor with resistance \( R \) as shown in the exercise, where \( I = \frac{V}{R} \). Grasping the concept of Ohm's Law is essential for anyone beginning to study electricity and electronics.

Cylindrical Conductor Properties

A cylindrical conductor possesses unique properties that are important in the field of electromagnetism. The conductor's shape affects its resistance and its ability to carry an electric current. The critical dimensions of a cylindrical conductor are its length \( L \) and its radius \( r \). The resistance of such a conductor is highly dependent on both dimensions, as reflected in the exercise solution where \( R = \rho \frac{L}{\pi r^2} \).

The cross-sectional area \( A = \pi r^2 \) of the cylinder plays a vital role in calculating the resistance using resistivity. It is equally important to note that cylindrical conductors are the shape of choice for most practical wires and cables due to their efficiency in conducting electric current and their ease of manufacture.

Vacuum Permittivity

The concept of vacuum permittivity, denoted as \( \epsilon_0 \), is fundamental in electromagnetism. It represents the capability of the vacuum of space to permit electric field lines. Vacuum permittivity is a constant with an approximate value of \( 8.854 \times 10^{-12} \text{F/m} \) (farads per meter) and represents the amount of opposition that an electric field faces when forming in a vacuum.

It comes into play when dealing with displacement currents in materials, as it is an integral part of the relationship between displacement current \( i_D \), the vacuum permittivity \( \epsilon_0 \), and the rate of change of the electric field or current as seen in the expression \( i_D = \epsilon_0 \frac{dI}{dt} \). This constant is crucial in the study of not just simple circuits, but also in more complex physics phenomena such as electromagnetic waves and their propagation in different media.