Study Guide

Electricity and Magnetism - Advanced Circuit Elements

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Advanced Circuit Elements

Real circuits use more than just resistors to manipulate charge. There's a whole armory of circuit elements electrical engineers have at their disposal, and like Q briefing James Bond, we're about to pass three of the more immediately useful ones onto you. No laser watch this time, but stick around for the sequel.

Real Batteries

First up: real batteries. While an idealized battery producing a voltage V is fine in theory, real batteries all have some internal resistance, r, that affects the circuit in which they're placed. The voltage V created by a real battery is:

Here,  is the emf created by the battery if it had no internal resistance, and I is the current provided by the battery to the circuit. Essentially, our real battery is a series combination of an ideal battery and a small resistor:

The AAs in your flashlight or Gameboy or, yes, even the Energizer bunny eventually die because r grows as the battery ages, until finally the voltage provided by the battery isn't enough to power much of anything.


Okay, so real batteries are kind of a letdown—like Q giving you a pen that's actually just a pen. And it's out of ink. But there are some circuit elements that live up to Bond's standards.

Take the capacitor. A capacitor is an object that quickly stores charge when a voltage is applied to it, and can quickly discharge when that voltage is removed. Camera flashes, stun guns, touch screens—these are all gadgets that depend on capacitance to function.

Capacitance (C) is a measure of how much charge can be stored between the parts of a capacitor when a voltage is applied to it. The simplest capacitor is a parallel plate capacitor, which consists of two square metal plates of area A separated by a distance d. A material called a dielectric can be inserted between the plates to change the configuration's capacitance if desired.

Looks like a blue mystery meat sandwich to us.

When attached to a battery, current flows around the circuit, but can't jump the distance between the plates. Instead, positive charge builds up on one plate, negative charge builds up on the other, and the current slows to a dribble.

The capacitance of a parallel plate capacitor with no dielectric is given by:

C is dependent on the size of the plates and their separation, but gives us a ratio of the charge stored on the plates versus the voltage applied to the capacitor by the battery:

Capacitance is measured in coulombs per volt, which we define as a farad (F).

When the battery is removed, the charge sits happily on the capacitor. But if we connect the two ends of the capacitor, say through a resistor:

All of a sudden the potential difference in the capacitor has a path to discharge. It will act like a battery with a voltage determined by C and q. Unlike a battery, it has the ability to move a lot of charge very, very quickly allowing circuits to light camera flashes. Or drop evil henchmen with 50,000 V of shaken not stirred.

Because capacitors are a way to store and release charge in a circuit, they are also a way to store and release energy. Charge building up on a capacitor will convert energy from the battery into stored energy in the capacitor, given by:

Just like resistors, we can add multiple capacitors to a circuit; however, they combine quite differently. If we attach two capacitors in parallel to a battery, they must have the same voltage drop across them, as given by Kirchhoff's Voltage Law. If these two capacitors have the same capacitance, the equation means that they must also both have an identical amount of charge build up on them. Then an equivalent capacitor to replace the parallel capacitors would have twice the charge on it for the same voltage drop—in other words, twice the capacitance. This is not at all what happens with resistors.

For capacitors in parallel and series, we have:

This is exactly the reverse of parallel and series combinations of resistors.


While capacitors store energy in the electric field between two plates, inductors store energy in the magnetic field of a coil of wire. Inductors are used in power transformers and electric motors, and so are crucial to many of 007's missions—the ejector seat of his Aston Martin wasn't ejecting anything without some sort of electric motor releasing the springs.

The voltage that appears across an inductor when it experiences a change in current Δ I is is based on its inductance (L):

This voltage will always be created in a direction that will try to resist the change in current, a consequence of Faraday's and Lenz's Laws.

How much energy an inductor can store is a combination of its inductance and the non-changing current through the coil:

Inductance, like capacitance, is dependent on the geometry of the device. It's measured in henries (H).

Inductors are combined in circuits in exactly the same way resistors are: inductances add normally in series and inversely in parallel.

And with that, you have all the tools for your current mission. While Q's armory is far from empty—there are still transistors, diodes, transducers, oscillators, resonators, logic gates, microphones, loudspeakers, motors, thermistors, lasers (obviously), and many more—those come into play in future installments. You wouldn't expect Bond to use the same old tricks every time, would you?

Common Mistakes

Resistors and inductors combine identically, but don't forget that capacitors are backwards—they combine in series like resistors combine in parallel, and vice versa.

Brain Snack

Touch screens are all based on the fact that human beings are just another circuit element. The screen of your smart phone is made of glass with a grid of transparent metal over it that conducts electricity. (You read that correctly—transparent metal. Mind = blown, right?) When you put your finger down on the screen, it changes the capacitance of that section of the grid, and the phone can detect exactly how far that catapult is supposed to launch the bird.

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