PHYS102: Introduction to Electromagnetism

3a. State Ohm's Law in words

• What happens when you apply a potential difference to the ends of a conductor? What physical quantity measures the rate of the flow of charge?
• What is the definition of the resistance of a conductor? What is the relationship between resistance, current, and potential difference between the ends of a conductor? When is this relationship called Ohm's Law?

If you put a conductor in an electric field (for example, if its ends are connected to a battery, so there is a voltage between each end), the free charges inside the conductor will begin to move. Current is the rate of the flow of charge. The ratio between the applied voltage (also called potential difference) and the current is defined as resistance: $R=\frac{V}{I}$. For some materials, this ratio is constant, and the conductor is said to obey Ohm's Law. These materials are known as Ohmic materials. In them, the potential difference between the ends of the conductor is proportional to the current through the conductor. Ohm's Law is $V=IR$.

Review this material in Ohm's Law.

3b. Apply Ohm's Law to simple circuits

• Sketch an example of a simple circuit containing a battery and a resistor. What is the relationship between the current in the circuit and the voltage supplied by the battery?

According to Ohm's Law, the current established in a circuit with one battery supplying a voltage $V$ is $I=\frac{V}{R}$. Here, $R$ can be the resistance of the only resistor in the circuit, or the equivalent resistance of the network of several resistance in the circuit.

Make sure you understand Example 20.4 in Ohm's Law: Resistance and Simple Circuits.

3c. Calculate effective resistance of a network of resistors

• Let's say you have two resistors. How can you connect them to a battery so they are connected in series? What is the relationship between the current through each resistor? What is the relationship of the potential differences between the ends of each resistor and the voltage supplied by the battery? Use these considerations to determine the equivalent resistance of two resistors connected in series.
• If you have two resistors, how can you connect them to a battery so they are connected in parallel? What is the relationship between the current through each resistor? What is the relationship of the potential differences between the ends of each resistor and the voltage supplied by the battery? Use these considerations to determine the equivalent resistance of two resistors connected in parallel.

When two or more resistors are connected in series:

• They have the same current going through them (this follows from the conservation of charge)

• The sum of the potential differences between the ends of each resistor equals the voltage supplied by the battery.

• The equivalent resistance is determined by the formula $R_{eq}=R_{1}+R_{2}+\cdots$.

When two or more resistors are connected in parallel:

• They have the same potential difference between the ends, which also equals the voltage supplied by the battery.

• The sum of currents through each resistor equals the total current in the circuit, or the current drawn from the battery.

• The equivalent resistance is determined by the formula $\frac{1}{R_{eq}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\cdots$.

Review this material in Resistors in Series and Parallel. Review a more advanced example in Combo Circuits.

3d. Determine the resistance of a cylindrical wire

• How do conducting materials resist the flow of a current, at the microscopic level?
• How does the resistance of a wire depend on the resistivity of the material and the length and cross-sectional area of the wire?

Resistivity is a characteristic of a conducting material and describes its ability to allow charges to flow. Resistivity depends on a variety of factors, including the density of atoms in the material and the material's temperature. The resistance of a wire made out of a material with resistivity $\rho$ is proportional to the wire's length (the longer the charges have to travel, the greater the resistance), and inversely proportional to its cross-sectional area (the greater the area, the more pathways for the charges to travel; hence, less resistance), and can be described with $R=\rho \frac{l}{A}$.

Review this material in Resistance and Resistivity.

3e. Compare and contrast voltage and current

• What conditions are necessary for a current to flow?
• What are some possible ways to create voltage?

Current is the amount of charge that passes through the cross-section of a conductor during a unit of time: $I=\frac{\Delta q}{\Delta t}$, or rate of flow of charge. Voltage is the potential difference between two points in space, or between the ends of a conductor. Current describes the motion of charged particles, whereas voltage measures the energy per unit of charge acquired or lost by these particles.

A good way to internalize the difference between current and voltage is to see how the two quantities are measured in practice. Review this in Ammeter and Voltmeter and DC Voltmeters and Ammeters.

3f. Use the junction and loop rules to analyze basic circuits

• What is the junction rule? Explain in terms of conservation of charge.
• What is the loop rule? Explain in terms of conservative forces and equipotential surfaces.

For some circuits, it is impossible to find an equivalent resistance network of resistors. We can use Kirchhoff's Rules (the Junction Rule and the Loop Rule) to determine the current that flows through each resistor in a circuit. We can apply these to any circuit.

The Junction Rule states that the sum of all currents entering a junction equals the sum of all currents leaving a junction.

The Loop Rule states that the algebraic sum of all changes in electric potential due to electromotive forces of the batteries and the voltage drops across the resistors equals zero for any closed loop of a circuit.

Review the drawings in Kirchhoff's Rules to understand these rules. Try to reproduce the drawings in the examples and make sure you can follow how they are labeled.

3g. Explain how a battery works

• Why is a battery a necessary component of an electrical circuit?
• What is an electromotive force?

A typical battery is an electrochemical cell. The chemical reaction inside the cell separates its positive and negative ions and makes them move in opposite directions. This results in a potential difference between the two ends of an electrical circuit. In this way, the battery converts chemical energy to electrical energy, and supplies that energy to the circuit. Electromotive force (EMF) is the amount of energy per unit of charge supplied by the battery. Despite the name, it is not a force, but rather work per unit charge, which is measured in volts. For an ideal battery with negligible internal resistance, electromotive force equals the output voltage.

Review batteries and electromotive force in Electromotive Force: Terminal Voltage.

3h. Calculate the power in a DC circuit

• How would you define power using the concepts of work and energy?
• What are the different ways to calculate the power dissipated by a resistor in DC (direct current) circuits in terms of current, resistance, and voltage?

In electromagnetism, we define power the same way as in mechanics: power is the rate at which work is performed or the rate at which energy is dissipated. As charges move through a conductor, they lose energy, which gets converted into thermal energy or light. The power dissipated by resistor R with current I going through it can be calculated as $P=I^{2}R$. Alternatively, since the voltage across the resistor is $V=IR$, the power can be calculated as $P=VI=\frac{V^{2}}{R}$.

Review the explanation for these formulas in Electric Power and Energy.

3i. Calculate the power in an AC circuit

• What is the difference between direct and alternating current, DC and AC?
• What is the difference between rms voltage and peak voltage?
• How does the average power delivered to a lightbulb relate to its resistance and the voltage?

Alternating current periodically changes direction. When connected to a lightbulb or some other resistor, this means the voltage must also change periodically, because Ohm's law states that current (I) and voltage ($V$) are proportional to each other. In a household AC supply, this periodic reversal occurs at a frequency of 50 or 60 Hertz. To determine how bright a lightbulb gets, you need to know the average power delivered to it, compounded over an entire cycle. Even though the current averages out to zero because it reverses periodically, the power does not average out to zero. This is because the power at every moment in time is $P=VI$, and if both reverse signs, then their product does not.

To calculate the average power, you have to use values for $V$ and $I$ that best represent the ever-changing currents and voltages. These are called the rms voltage ($V_{rms}$) and rms current ($I_{rms}$). Here, "rms" stands for "root-mean-square", which describes a way of forming the average where negative signs are dropped so that they do not lead to everything canceling out to zero. They are related to the peak values $V_{0}$ and $I_{0}$ by

$V_{rms}=\frac{V_{0}}{\sqrt{2}}$ and $I_{rms}=\frac{I_{0}}{\sqrt{2}}$.

Then the average power is $P_{ave}=V_{rms}I{rms}$

Review Alternating Current vs. Direct Current for examples of how to calculate the rms values of current and voltage, and how to obtain the average power.

Unit 3 Vocabulary

You should be familiar with these terms to complete the final exam.

• average power
• battery
• current
• Electromotive Force (EMF)
• Kirchhoff's Rules: Junction Rule and Loop Rule
• Ohm's Law
• parallel connection
• power
• resistance
• resistivity
• resistor
• rms voltage and current
• series connection
• voltage