Question

A wire of radius \(1.0 \mathrm{~mm}\) carries a current of 20.0 A. The wire is connected to a parallel plate capacitor with circular plates of radius \(R=4.0 \mathrm{~cm}\) and a separation between the plates of \(s=2.0 \mathrm{~mm} .\) What is the magnitude of the magnetic field due to the changing electric field at a point that is a radial distance of \(r=1.0 \mathrm{~cm}\) from the center of the parallel plates? Neglect edge effects.

Short answer

Answer: The magnitude of the magnetic field B due to the changing electric field at a point that is a radial distance of r = 1.0 cm from the center of the parallel plates is approximately 1.13 × 10^{-9} T.

Step-by-step solution

Compute the displacement current

To calculate the displacement current in the problem, we need to first find the change in electric flux with respect to time (dΦ/dt). The electric field E in the capacitor is given by the formula: E = V/s, where V is the voltage across the capacitor and s is the separation between the plates. If the current I is flowing through the wire, we can relate the rate of change in the charge Q of the capacitor to this current by dQ/dt=I. Since Q = CV, where C is the capacitance of the capacitor, we get dQ/dt = C*dV/dt. The capacitance C of the parallel plate capacitor is given by C = ε₀ * A/s, where ε₀ is the permittivity of free space, A is the area of one circular plate, and s is the separation between the plates. So, the displacement current I_d is given by I_d = ε₀ * (dΦ/dt) = ε₀ * A * (dE/dt).

Calculate the magnetic field using the Ampère-Maxwell law

Now we can use the Ampère-Maxwell law to find the magnetic field B at a radial distance of r = 1.0 cm from the center of the plates. According to the Ampère-Maxwell law, ∮ B dl = μ₀ * (I_enclosed + I_displacement), where μ₀ is the permeability of free space, I_enclosed is the enclosed current in the loop, and I_displacement is the displacement current through the loop. Considering a circular loop of radius r=1.0 cm along the radial distance from the center of the parallel plates, the enclosed current I_enclosed = 0, as we are not considering any wire inside the loop. The displacement current through the loop I_displacement = ε₀ * A_p * (dE/dt), where A_p = πr² is the area of the circular loop of radius r. Then, the Ampère-Maxwell law can be written as: ∮B dl = μ₀ * ε₀ * A_p * (dE/dt), which, considering that the magnetic field is tangential to the loop and constant along it, becomes B * 2πr = μ₀ * ε₀ * πr² * (dE/dt).

Solve for the magnetic field B

Finally, we can solve the equation for B: B = (μ₀ * ε₀ * r * (dE/dt))/(2π) Now, we need to find (dE/dt) to get the magnitude of the magnetic field B. We have E = V/s, so dE/dt = (dV/dt) / s. From Step 1, we know that dQ/dt = C*dV/dt = I, so dV/dt = I/C. So, dE/dt = I/(Cs). Now, plug in the values: r = 1.0 cm = 0.01 m, I = 20 A, ε₀ = 8.85 × 10^{-12} F/m, μ₀ = 4π × 10^{-7} Tm/A, A = π*(0.04 m)², and s = 2.0 mm = 0.002 m. First calculate the capacitance C = ε₀ * A/s, and then substitute the values for dE/dt and B. After substituting the values and performing the calculations, the magnitude of the magnetic field B due to the changing electric field at a point that is a radial distance of r = 1.0 cm from the center of the parallel plates is approximately 1.13 × 10^{-9} T.

Key concepts

Displacement Current

Understanding the concept of displacement current is essential in explaining how electric fields can produce magnetic fields, even in the absence of conventional current. In classical electromagnetism, the displacement current is not an actual movement of charge; instead, it represents a time-varying electric field within a capacitor during charging or discharging.

The idea was introduced by James Clerk Maxwell, who realized that an inconsistency existed within Ampère's law, which describes how an electric current generates a magnetic field. Maxwell noted that a changing electric field between the plates of a capacitor should also produce a magnetic field, similar to the effect of a real current moving through a wire. To account for this, he introduced the term displacement current, mathematically described as \( I_d = \varepsilon_0 \frac{d\Phi_E}{dt} \), where \( I_d \) is the displacement current, \( \varepsilon_0 \) is the permittivity of free space, and \( \frac{d\Phi_E}{dt} \) is the rate of change of electric flux \( \Phi_E \).

In our exercise, as the electric field changes between the capacitor's plates, a displacement current is generated, and this provides a complete path for magnetic field lines, thereby maintaining the consistency of Maxwell's equations.

Ampère-Maxwell Law

The Ampère-Maxwell law is a cornerstone in our understanding of how electric currents and changing electric fields contribute to the magnetic field. It extends Ampère's Circuital Law, which originally stated that the line integral of the magnetic field \( \mathbf{B} \) around a closed loop is proportional to the current passing through the loop. Maxwell's addition, the displacement current, allows us to include the effects of changing electric fields.

The full Ampère-Maxwell law is described by the equation \( \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left( I_{\text{enclosed}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right) \), where \( \oint \mathbf{B} \cdot d\mathbf{l} \) represents the line integral of the magnetic field around a closed path, \( \mu_0 \) is the permeability of free space, and \( I_{\text{enclosed}} \) is the conduction current enclosed by the path. The term \( \varepsilon_0 \frac{d\Phi_E}{dt} \) represents the displacement current due to the changing electric flux. This law shows that both conduction current and displacement current can generate a magnetic field, and it is used in our exercise to calculate the magnetic field at a given point due to the changing electric field in the capacitor.

Parallel Plate Capacitor

A parallel plate capacitor is a simple yet fundamental electrical component that stores energy in the form of an electric field between its two conductive plates, separated by an insulating material (or vacuum). The capacitance, \( C \), is a measure of the ability of the capacitor to store charge, and it depends on the geometry of the plates and the permittivity of the insulating material.

The capacitance is given by the formula \( C = \varepsilon_0 \frac{A}{s} \), where \( A \) is the area of one of the plates, \( s \) is the separation between the plates, and \( \varepsilon_0 \) is the permittivity of free space. When a voltage \( V \) is applied across the plates, an electric field \( E \) is established, expressed as \( E = \frac{V}{s} \).

In our exercise scenario, the circular plates create a capacitor that is being charged, resulting in the establishment of a displacement current. This capacitor's ability to store energy is crucial to the generation of the magnetic field investigated in the problem.

Electric Flux

Electric flux, denoted by \( \Phi_E \), is a measure of the quantity of electric field penetrating through a given area. It is calculated by the product of the electric field \( \mathbf{E} \) and the area \( A \) through which the field lines pass, mathematically expressed as \( \Phi_E = \mathbf{E} \cdot \mathbf{A} \) if the electric field is uniformly distributed and perpendicular to the area.

For non-uniform or non-perpendicular electric fields, we use the dot product to calculate electric flux: \( \Phi_E = \mathbf{E} \cdot \mathbf{A} \cos\theta \), where \( \theta \) is the angle between the electric field lines and the normal (perpendicular) to the area \( A \). Changes in electric flux over time in the presence of changing electric fields, like in the capacitor in our exercise, lead to the creation of displacement currents, which are crucial in the generation of magnetic fields through the Ampère-Maxwell law.