Resistors, Inductors and Capacitors in Electric Circuits

Resistors, inductors, and capacitors are fundamental passive elements in electric circuits. Understanding their physical meaning and governing equations is essential for circuit analysis.

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Resistors

Resistance is the physical property of an element that opposes the flow of electric current. It is represented by the symbol $R$.

For a uniform wire:

$$R = \rho \frac{l}{A}$$

where:

• $\rho$ = resistivity
• $l$ = length of the conductor
• $A$ = cross-sectional area

The unit of resistance is the ohm (Ω).

A conductor has a resistance of 1 Ω if a current of 1 A flows when 1 V is applied across its terminals.

Ohm’s Law

$$v = iR$$

where:

• $v$ = voltage
• $i$ = current
• $R$ = resistance

Ohm’s law states that current is directly proportional to voltage, provided temperature and physical conditions remain constant.

Limitations of Ohm’s Law

• It is empirical and not universal.
• Some materials are non-ohmic.
• Resistivity varies with temperature.

Power in a Resistor

$$p = vi = i^2 R = \frac{v^2}{R}$$

Since $i^2 \ge 0$, the power absorbed is always positive.
Thus, a resistor is a passive element.

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Inductors

An inductor produces a voltage proportional to the rate of change of current:

$$v = L \frac{di}{dt}$$

where:

• $L$ = inductance (Henry)

Energy Stored in an Inductor

Power:

$$p = vi = L i \frac{di}{dt}$$

Energy stored:

$$w(t) = \frac{1}{2} L i^2(t)$$

Since $w(t) \ge 0$, the inductor stores energy in a magnetic field and is a passive element.

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Capacitors

A capacitor consists of two conducting plates separated by a dielectric.

Capacitance is defined as:

$$C = \frac{q}{v}$$

Current–Voltage Relation

$$i = C \frac{dv}{dt}$$

Energy Stored in a Capacitor

$$w(t) = \frac{1}{2} C v^2(t)$$

Since $w(t) \ge 0$, the capacitor stores energy in an electric field and is also a passive element.

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Summary

• A resistor dissipates energy.
• An inductor stores energy in a magnetic field.
• A capacitor stores energy in an electric field.

All three are classified as passive elements because they do not generate energy but either dissipate or store it.