Imagine that a steady current I flows in a straight cylindrical wire of radius R₀ and resistivity ρ.
(a) If the current is then changed at a rate dI/dt, show that a displacement current ID exists in the wire of magnitude ε₀ρ(dI/dt).
(b) If the current in a copper wire is changed at the rate of 1.0 A/ms, determine the magnitude of ID.
(c) Determine the magnitude of the magnetic field BD created by ID at the surface of a copper wire with R₀ = 1.00 mm. Compare (as a ratio) BD with the field created at the surface of the wire by a steady current of 1.0 A.

Verified step by step guidance

1. Step 1: Begin by understanding the concept of displacement current. Displacement current arises in situations where the electric field changes with time, and it is given by the formula: $I{D}_{}=\epsilon {₀}_{}\rho \left(\frac{d}{I}\right)$. This formula relates the displacement current to the rate of change of the actual current in the wire.
2. Step 2: For part (a), derive the expression for the displacement current. The resistivity $\rho$ of the wire and the permittivity of free space $\epsilon {₀}_{}$ are constants. The displacement current $I{D}_{}$ is proportional to the rate of change of the actual current $\frac{d}{I}$. Substitute these values into the formula to show that $I{D}_{}=\epsilon {₀}_{}\rho \left(\frac{d}{I}\right)$.
3. Step 3: For part (b), calculate the magnitude of the displacement current $I{D}_{}$ when the rate of change of current $\frac{d}{I}$ is given as 1.0 A/ms. Use the formula derived in part (a) and substitute the values of $\epsilon {₀}_{}$ (8.85 × 10⁻¹² F/m), $\rho$ (resistivity of copper, approximately 1.7 × 10⁻⁸ Ω·m), and $\frac{d}{I}$ (1.0 A/ms).
4. Step 4: For part (c), determine the magnetic field $B{D}_{}$ created by the displacement current at the surface of the wire. Use Ampere's law, which relates the magnetic field to the current enclosed: $B=\frac{\mu }{{₀}_{}}/\left(2\pi R\right)$, where $\mu {₀}_{}$ is the permeability of free space (4π × 10⁻⁷ T·m/A), $I$ is the displacement current, and $R$ is the radius of the wire.
5. Step 5: Compare the magnetic field $B{D}_{}$ created by the displacement current to the magnetic field created by a steady current of 1.0 A. Use the same formula for the magnetic field, substituting the steady current value instead of the displacement current. Calculate the ratio $\frac{B}{D_{}}$ to compare the two fields.

Key Concepts

Displacement Current

Displacement current is a concept introduced by James Clerk Maxwell to account for changing electric fields in situations where traditional current does not flow. It is defined as ε₀(dΦ_E/dt), where ε₀ is the permittivity of free space and dΦ_E/dt is the rate of change of electric flux. In the context of a wire with changing current, the displacement current can be expressed in terms of resistivity and the rate of change of current, illustrating how electric fields can influence magnetic fields even in the absence of physical charge movement.

Resistivity

Resistivity is a material property that quantifies how strongly a given material opposes the flow of electric current. It is denoted by the symbol ρ and is measured in ohm-meters (Ω·m). The resistivity of a material affects the amount of current that can flow through it for a given voltage, and it plays a crucial role in determining the displacement current when the actual current changes over time.

Magnetic Field due to Current

The magnetic field generated by an electric current is described by Ampère's Law, which states that the magnetic field around a conductor is proportional to the current flowing through it. For a straight cylindrical wire, the magnetic field at the surface can be calculated using the formula B = (μ₀I)/(2πR), where μ₀ is the permeability of free space, I is the current, and R is the distance from the wire's center. Understanding this relationship is essential for comparing the magnetic fields produced by both steady and displacement currents.