Show that the quantity ϵ0(dΦe/dt) has units of current.

Ch 31: Electromagnetic Fields and Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 31: Electromagnetic Fields and Waves

문제 9

31장, 문제 9

Show that the quantity ϵ₀(dΦ_e/dt) has units of current.

검증된 단계별 안내

Step 1: Begin by identifying the physical quantities involved. Here, ϵ₀ represents the permittivity of free space, dΦₑ/dt is the rate of change of electric flux, and we need to show that the product ϵ₀(dΦₑ/dt) has units of current.

Step 2: Recall the SI unit of permittivity of free space (ϵ₀). It is measured in farads per meter (F/m). A farad (F) is equivalent to coulombs per volt (C/V), so ϵ₀ has units of C/(V·m).

Step 3: Consider the electric flux Φₑ. Electric flux is defined as the integral of the electric field over an area, Φₑ = ∫E·dA. The electric field (E) has units of volts per meter (V/m), and area (A) has units of square meters (m²). Therefore, Φₑ has units of V·m.

Step 4: The rate of change of electric flux, dΦₑ/dt, involves dividing the units of Φₑ (V·m) by time (seconds, s). Thus, dΦₑ/dt has units of V·m/s.

Step 5: Multiply ϵ₀ (C/(V·m)) by dΦₑ/dt (V·m/s). The units become (C/(V·m)) × (V·m/s) = C/s. Since coulombs per second (C/s) is the unit of electric current, this demonstrates that ϵ₀(dΦₑ/dt) has units of current.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Electric Flux (Φe)

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, taking into account the angle between the field lines and the normal to the surface. The unit of electric flux is the volt-meter (V·m), which can also be expressed in terms of base SI units as kg·m²/(s³·A).

Permittivity of Free Space (ϵ0)

The permittivity of free space (ϵ0) is a fundamental physical constant that characterizes how electric fields interact with the vacuum of space. It has a value of approximately 8.85 x 10^-12 F/m (farads per meter) and is crucial in determining the capacitance of capacitors and the behavior of electric fields in free space. This constant plays a key role in Maxwell's equations, which describe electromagnetism.

Current (I)

Electric current (I) is the flow of electric charge through a conductor, typically measured in amperes (A). One ampere is defined as one coulomb of charge passing through a point in a circuit per second. In the context of the equation, showing that ϵ0(dΦe/dt) has units of current involves demonstrating that the units of the expression correspond to those of amperes, linking electric flux change over time to charge flow.