Ch. 31 - Maxwell's Equations and Electromagnetic Waves

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 31 - Maxwell's Equations and Electromagnetic Waves

Problem 66

Chapter 30, Problem 66

Show that displacement current, ε₀ (dΦE/dt), has the SI units of amperes.

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Start by recalling the formula for displacement current: \( I_d = \varepsilon_0 \frac{d\Phi_E}{dt} \), where \( \varepsilon_0 \) is the permittivity of free space, and \( \frac{d\Phi_E}{dt} \) is the rate of change of electric flux.

The SI unit of \( \varepsilon_0 \) (permittivity of free space) is \( \text{F/m} \) (farads per meter). A farad (F) is defined as \( \text{C/V} \) (coulombs per volt). Therefore, \( \varepsilon_0 \) has units of \( \text{C/(V·m)} \).

The electric flux \( \Phi_E \) is defined as \( \Phi_E = E \cdot A \), where \( E \) is the electric field (units: \( \text{V/m} \)) and \( A \) is the area (units: \( \text{m}^2 \)). Thus, \( \Phi_E \) has units of \( \text{V·m} \).

The rate of change of electric flux, \( \frac{d\Phi_E}{dt} \), has units of \( \text{V·m/s} \).

Now, multiply \( \varepsilon_0 \) (units: \( \text{C/(V·m)} \)) by \( \frac{d\Phi_E}{dt} \) (units: \( \text{V·m/s} \)). The \( \text{V} \) and \( \text{m} \) terms cancel appropriately, leaving \( \text{C/s} \), which is the SI unit of current (amperes). Thus, \( I_d \) has units of amperes.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Displacement Current

Displacement current is a term introduced by James Clerk Maxwell to account for changing electric fields in situations where there is no actual flow of charge, such as in capacitors. It is defined mathematically as ε₀ (dΦE/dt), where ε₀ is the permittivity of free space and dΦE/dt is the rate of change of electric flux. This concept is crucial for understanding how electric and magnetic fields interact in electromagnetic theory.

Electric Flux

Electric flux (ΦE) is a measure of the electric field passing through a given area. It is calculated as the product of the electric field (E) and the area (A) through which it passes, adjusted for the angle between the field lines and the normal to the surface. The unit of electric flux is volt-meters (V·m), which is equivalent to newton-meters squared per coulomb (N·m²/C), and it plays a key role in Faraday's law of electromagnetic induction.

SI Units of Current

The SI unit of electric current is the ampere (A), which is defined as the flow of one coulomb of charge per second. In the context of displacement current, showing that ε₀ (dΦE/dt) has the units of amperes involves analyzing the units of each component: ε₀ has units of farads per meter (F/m), and dΦE/dt has units of volts-meters per second (V·m/s). Through dimensional analysis, it can be demonstrated that the resulting units simplify to amperes.

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