Physics (9702) | Potential difference and power

Welcome to Potential Difference and Power!

Hello future physicists! This chapter is absolutely fundamental to understanding how electricity works in the real world—from powering your phone to heating your kettle. We are moving beyond just the flow of current and starting to look at the energy transfers happening in a circuit.

Don't worry if electricity seems a bit abstract. We'll use simple analogies to break down the concepts of Potential Difference (Voltage) and Electric Power into digestible pieces. By the end, you will be a master of the key electrical formulas!

1. Understanding Potential Difference (p.d.)

1.1 Defining Potential Difference (Voltage)

Imagine an electric circuit as a water slide. For the water (charge) to move, it needs a height difference. In electricity, this "height difference" or driving force is called the Potential Difference (p.d.), or simply Voltage (V).

The formal definition is about energy:

Potential difference (V) across a component is defined as the energy transferred per unit charge when charge passes through the component.

Analogy: The Electric Hill

Think of a charge (Q) moving through a resistor. It loses electrical potential energy and converts it into heat and light. The potential difference (V) tells you exactly how much energy (W) is lost by every coulomb of charge (Q) passing through that component.

1.2 The P.D. Formula

Mathematically, potential difference is defined by the relationship between energy (work done) and charge:

\[V = \frac{W}{Q}\]

Where:

• \(V\) is the potential difference, measured in Volts (V).
• \(W\) is the work done or energy transferred, measured in Joules (J).
• \(Q\) is the charge passing through, measured in Coulombs (C).

Key Point: 1 Volt (V) is equivalent to 1 Joule of energy transferred per 1 Coulomb of charge ($1 \, \text{V} = 1 \, \text{J/C}$).

Key Takeaway for Section 1: P.D. is the electrical 'push' or 'energy loss' per unit of charge. $V = W/Q$ is your fundamental link between electricity and energy.

2. Electric Power (P) and Energy Transfer

Power is a concept you've met before (Chapter 5: Work, Energy and Power). It is the rate at which energy is transferred. In electrical circuits, we want to know how fast the electrical energy is being converted into other forms (like heat, light, or mechanical energy).

2.1 Defining Electric Power

Electric Power (P) is the rate at which electrical energy is transferred or dissipated in a component.

The general formula for power is:

\[P = \frac{W}{t}\]

Where:

• \(P\) is the power, measured in Watts (W).
• \(W\) is the energy transferred (Joules).
• \(t\) is the time taken (seconds).

2.2 The Primary Electrical Power Formula: P = VI

This is one of the most important relationships in electricity. We can derive it by combining the definitions of P.D. and current.

Step 1: Start with the P.D. definition: \[W = VQ\]

Step 2: Substitute this into the Power definition: \[P = \frac{W}{t} = \frac{VQ}{t}\]

Step 3: Recognize that Current $I = Q/t$: \[P = V \times \left(\frac{Q}{t}\right) = VI\]

The result is:

\[P = VI\]

Key Takeaway for Section 2: Electrical power is simply the voltage multiplied by the current. $P = VI$ is your starting point for all electrical power calculations.

3. Power Formulas Involving Resistance (R)

Often, a problem will give you resistance (R) but not voltage (V) or current (I). We can use Ohm's Law ($V = IR$) to substitute and find two alternative power formulas.

3.1 The Three Power Formulas

Formula 1: Substituting V

If we know $I$ and $R$, but not $V$, substitute $V = IR$ into the primary equation $P=VI$:

\[P = (IR)I\]

\[P = I^2R\]

This formula is particularly useful when calculating energy dissipated as heat (often called Joule heating), as it only depends on the current flowing and the resistance.

Formula 2: Substituting I

If we know $V$ and $R$, but not $I$, rearrange Ohm's law to get $I = V/R$, then substitute this into $P=VI$:

\[P = V \left(\frac{V}{R}\right)\]

\[P = \frac{V^2}{R}\]

3.2 When to Use Which Formula?

Choosing the right formula can save you a step or two in calculations.

• Use $P = VI$: When dealing with power sources (batteries) or simple total circuit power, and you know the voltage applied and total current drawn.
• Use $P = I^2R$: When analyzing circuits in series. Current (I) is constant in series circuits, making this formula ideal for comparing power dissipation in different resistors. (Great for heating calculations!)
• Use $P = V^2/R$: When analyzing circuits in parallel. Voltage (V) is constant across parallel components, making this formula ideal for comparing power dissipation.

Common Mistake Alert!
Students often confuse $P = I^2R$ and $P = V^2/R$. Remember the context: if $I$ is constant (Series), use $I^2R$. If $V$ is constant (Parallel), use $V^2/R$.

Key Takeaway for Section 3: You have three powerful tools for calculating electrical power: $P=VI$, $P=I^2R$, and $P=V^2/R$. Use Ohm's Law to switch between them!

4. Electromotive Force (e.m.f.) and Internal Resistance

When we talk about a battery or power source, the total energy it supplies isn't simply called P.D.; it's called Electromotive Force (e.m.f.), symbolized by $\mathcal{E}$.

4.1 Defining E.M.F. ($\mathcal{E}$)

The definition of e.m.f. is very similar to P.D., but it describes the source, not a component:

Electromotive Force ($\mathcal{E}$) is defined as the energy transferred per unit charge in driving charge around a complete circuit.

It has the same unit as potential difference (Volts, V), and the formula is formally the same: $\mathcal{E} = W/Q$, where W is the total energy supplied by the source.

Distinguishing E.M.F. ($\mathcal{E}$) and P.D. (V)

This distinction is crucial for exam success.

• EMF ($\mathcal{E}$): The total electrical energy the source generates per coulomb of charge. (The 'promised' voltage).
• P.D. (V): The energy dissipated or transferred per coulomb of charge *outside* the source (across external components). (The 'delivered' voltage).

Analogy: If EMF is the total salary you earn, P.D. is the amount left after taxes (internal resistance) have been deducted.

4.2 The Effect of Internal Resistance (r)

All power sources (batteries, generators) have some internal opposition to the flow of charge. This is called internal resistance (r).

When current (I) flows, some of the energy generated by the source is wasted internally (dissipated as heat inside the battery). This energy loss is the lost volts ($v$).

According to Ohm's law, the lost volts are calculated by:

\[v = Ir\]

4.3 Terminal Potential Difference

The voltage you measure across the terminals of the source when current is flowing is the Terminal Potential Difference ($V_T$). It is the EMF minus the lost volts.

This relationship is given by:

\[\mathcal{E} = V_T + Ir\]

Where $V_T$ is the P.D. across the external load resistance ($R$), so $V_T = IR$.

Substituting $V_T = IR$ into the equation gives the full circuit expression:

\[\mathcal{E} = IR + Ir = I(R+r)\]

Key Takeaway for Section 4: EMF is the total energy supplied by the source ($\mathcal{E}$), while the terminal P.D. ($V_T$) is the usable voltage. The difference is the voltage lost due to internal resistance ($Ir$).

5. Review of Key Formulas

Here is a summary of the essential equations you must recall and use:

You've tackled the core concepts of electrical energy transfer! Mastering these definitions and three power formulas is crucial for success in circuits. Keep practicing substituting V=IR into P=VI—this is where your problem-solving skills shine!