Welcome to the Basics of Electricity!
Hello future physicists! This chapter is your foundation for understanding how electricity works, from the microscopic movement of charge to complex circuits that power our lives. Don't worry if you find circuit diagrams tricky—we'll break down the concepts of charge flow, voltage, and resistance using simple analogies. Mastering these basics (Section 3.4 in your syllabus) is crucial for success in all areas of electromagnetism!
3.4.1 Defining the Core Concepts: Current, PD, and Resistance
Electric Current \((I)\): The Flow Rate of Charge
Imagine electricity like water flowing in a pipe. The current is simply how much water (charge) passes a specific point every second.
Electric current (\(I\)) is defined as the rate of flow of charge.
- Formula: $$\mathbf{I = \frac{\Delta Q}{\Delta t}}$$ Where \(\Delta Q\) is the change in charge (in Coulombs, C) passing a point in time \(\Delta t\) (in seconds, s).
- Unit: The SI unit for current is the Ampere (A). One Ampere is equivalent to one Coulomb per second (1 A = 1 C s\(^{-1}\)).
- Did you know? In metals, the charge carriers are electrons, which flow from negative to positive. However, by convention, we define conventional current as flowing from positive (+) to negative (-).
Potential Difference \((V)\) (Voltage): Energy per Charge
If current is the flow, potential difference (PD) is the "push" required to make it flow. Think of it as the energy required to move the charge.
Potential difference (\(V\)), or voltage, between two points is the work done (\(W\)) per unit charge (\(Q\)) to move a positive test charge between those points.
- Formula: $$\mathbf{V = \frac{W}{Q}}$$ Where \(W\) is the work done or energy transferred (in Joules, J).
- Unit: The SI unit for PD is the Volt (V). One Volt is equivalent to one Joule per Coulomb (1 V = 1 J C\(^{-1}\)).
- Analogy: If current is the amount of water, PD is the pressure difference driving that water through the pipe.
Resistance \((R)\): The Opposition to Flow
When charges move through a conductor, they collide with atoms and other charges, losing energy as heat. This opposition to current flow is called resistance.
Resistance (\(R\)) is defined as the ratio of potential difference (\(V\)) across a component to the current (\(I\)) flowing through it.
- Formula: $$\mathbf{R = \frac{V}{I}}$$
- Unit: The SI unit for resistance is the Ohm (\(\Omega\)).
Quick Review: Basics of Electricity
- Current \(I = \Delta Q / \Delta t\) (Amperes, A)
- Potential Difference \(V = W / Q\) (Volts, V)
- Resistance \(R = V / I\) (Ohms, \(\Omega\))
3.4.2 Current-Voltage Characteristics and Ohm's Law
Ohm's Law: A Special Case
Many components obey a simple rule established by George Ohm.
Ohm's Law states that the current flowing through a metallic conductor is directly proportional to the potential difference across its ends, provided that physical conditions (like temperature) remain constant.
Mathematically, this means \(I \propto V\) (or \(V \propto I\)). The constant of proportionality is the resistance \(R\).
Current-Voltage (\(I-V\)) Characteristics
The \(I-V\) characteristic is a graph showing how the current through a component changes as the PD across it changes. The gradient of this graph is related to \(1/R\). (Remember: questions may plot \(I\) on the x-axis or the y-axis).
1. Ohmic Conductor (e.g., Fixed Resistor at Constant Temperature)
- Characteristic: The graph is a straight line passing through the origin.
- Resistance: The resistance \(R\) is constant because \(V/I\) is constant (Ohm's law is obeyed).
2. Filament Lamp (Non-Ohmic)
- Characteristic: The graph is a curve where the gradient decreases as \(V\) and \(I\) increase.
- Why it's non-ohmic: As current flows, the filament gets hotter. Increased temperature causes the metal lattice ions to vibrate more vigorously. This increases the frequency of collisions with moving electrons, thus increasing the resistance.
3. Semiconductor Diode (Non-Ohmic)
- Characteristic: Current flows easily in one direction (forward bias) after a certain "threshold" voltage (around 0.6 V), but almost zero current flows in the opposite direction (reverse bias).
- Resistance: Resistance is very high in reverse bias and very low (almost zero) in forward bias after the threshold is reached.
Ideal Meters
When measuring current and voltage in a circuit, we assume our meters do not interfere with the circuit's operation:
- An Ideal Ammeter (measures current) is placed in series and has zero resistance.
- An Ideal Voltmeter (measures PD) is placed in parallel and has infinite resistance.
(Why infinite/zero? We want the ammeter to allow all the current through, and the voltmeter to draw zero current itself.)
Key Takeaway: Ohmic vs. Non-Ohmic
The definition of resistance is always \(R=V/I\). Ohm's Law only applies if R is constant (straight line I-V graph).
3.4.3 Resistivity (\(\rho\)) and Material Properties
Resistance (\(R\)) depends on the shape of the conductor (its length and cross-sectional area) and the material it is made of.
- Resistance is proportional to length (\(L\)). (Longer wire = more collisions).
- Resistance is inversely proportional to cross-sectional area (\(A\)). (Wider wire = more pathways for current).
To compare materials independent of shape, we use resistivity.
Resistivity (\(\rho\)) is a property of the material itself, defined by the relationship:
- Formula: $$\mathbf{\rho = \frac{RA}{L}}$$
- Unit: The SI unit for resistivity is the Ohm-metre (\(\Omega\) m).
The Effect of Temperature on Resistance
Resistance is not always constant, especially when temperature changes:
- Metal Conductors: As temperature increases, the resistance of metals increases. (As seen in the filament lamp characteristic).
-
NTC Thermistors (Negative Temperature Coefficient): These are semiconductors whose resistance decreases significantly as temperature increases.
(Memory Aid: NTC = Negative Temperature Coefficient, meaning R and T move in opposite directions.)
Application: Thermistors are highly effective as temperature sensors in circuits, for example, in refrigerators or monitoring engine temperatures.
Superconductivity
Under extremely cold conditions, some materials exhibit a remarkable property called superconductivity.
- A superconductor has zero resistivity below a specific critical temperature.
- If there is zero resistance, there is zero energy loss due to heating.
-
Applications:
- Producing extremely strong magnetic fields (used in MRI scanners or particle accelerators).
- Significantly reducing energy loss in the transmission of electric power (as P = I\(^2\)R, if R=0, power loss is zero).
Key Takeaway: Resistivity
Resistivity (\(\rho\)) is the material constant. For metals, heating increases resistance; for NTC thermistors, heating decreases resistance.
3.4.4 Electrical Circuits, Energy, and Power
Power and Energy in DC Circuits
Since voltage is energy per unit charge (\(V=W/Q\)) and current is charge per unit time (\(I=\Delta Q/\Delta t\)), the energy delivered or dissipated (\(E\) or \(W\)) and power (\(P\)) can be calculated:
- Energy transferred (\(E\)): $$\mathbf{E = IVt}$$
- Power (\(P\)): (Power is the rate of energy transfer, \(P = E/t\)) $$\mathbf{P = IV}$$
- Alternative Power Formulas (using \(R=V/I\)): $$\mathbf{P = I^2R = \frac{V^2}{R}}$$
- Unit: Power is measured in Watts (W). Energy is measured in Joules (J).
Conservation Laws in DC Circuits
These fundamental laws must be obeyed in all circuits:
- Conservation of Charge: Charge cannot be created or destroyed. In a circuit, the total current entering a junction must equal the total current leaving it. (This is Kirchhoff's First Law, though you only need to know the principle).
- Conservation of Energy: Energy cannot be created or destroyed. In a closed loop, the total energy supplied by the source must equal the total energy dissipated in all the components. (This relates to Kirchhoff's Second Law).
Combining Resistors
Resistors are combined in two basic ways:
Resistors in Series
Components are connected end-to-end, forming a single path for the current.
- Current (\(I\)): The current is the same through every component: \(I_{total} = I_1 = I_2 = I_3\).
- Voltage (\(V\)): The voltage is shared across the components: \(V_{total} = V_1 + V_2 + V_3\).
- Total Resistance (\(R_T\)): $$\mathbf{R_{T} = R_1 + R_2 + R_3 + ...}$$ (The total resistance is always greater than the largest individual resistor.)
Resistors in Parallel
Components are connected across the same two points, providing alternative paths for the current.
- Current (\(I\)): The current splits between the paths: \(I_{total} = I_1 + I_2 + I_3\).
- Voltage (\(V\)): The voltage is the same across every parallel branch: \(V_{total} = V_1 = V_2 = V_3\).
- Total Resistance (\(R_T\)): $$\mathbf{\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...}$$ (The total resistance is always less than the smallest individual resistor.)
Common Mistake Alert!
When calculating parallel resistance, remember that the result of the reciprocal sum gives you \(\frac{1}{R_T}\). You must take the final reciprocal to find \(R_T\) itself!
3.4.5 The Potential Divider
A potential divider is a simple circuit consisting of two or more resistors (or components) connected in series across a power supply. Its purpose is to divide the potential difference (voltage) of the supply between the components.
Function
A potential divider allows you to supply a constant or variable potential difference that is less than the supply voltage.
The voltage across any single component in the series circuit is proportional to its resistance, relative to the total resistance.
For two resistors \(R_1\) and \(R_2\) in series connected to supply voltage \(V_{in}\), the output voltage \(V_{out}\) across \(R_2\) is: $$\mathbf{V_{out} = V_{in} \left( \frac{R_2}{R_1 + R_2} \right)}$$
Potential Dividers as Sensors
We can replace one of the fixed resistors in a potential divider with a sensor component (like a variable resistor, thermistor, or LDR) to create a voltage output that responds to environmental changes.
- Variable Resistor (Rheostat): By adjusting the variable resistor, the ratio of resistance changes, allowing the output PD to be varied manually.
- Thermistor (NTC): Used for temperature sensing. If the temperature increases, the thermistor's resistance decreases (for an NTC type), causing the voltage across it to fall (or the voltage across the fixed resistor to rise).
- Light Dependent Resistor (LDR): Used for light sensing. When light intensity increases, the LDR's resistance decreases. This affects the voltage ratio in the potential divider circuit.
Quick Review: Potential Dividers
A potential divider splits the voltage. By using variable components (like LDRs or thermistors), it converts a physical change (light, temperature) into a measurable voltage change.
3.4.6 Electromotive Force (e.m.f.) and Internal Resistance
All power sources (like batteries) are not perfectly ideal. They have their own internal resistance, which affects the voltage delivered to the external circuit.
Electromotive Force (\(\mathcal{E}\))
Electromotive force (\(\mathcal{E}\)) is the total energy (\(E\)) supplied by the source per unit charge (\(Q\)) passing through it.
$$\mathbf{\mathcal{E} = \frac{E}{Q}}$$The e.m.f. is the maximum potential difference the source can provide, measured when no current is flowing (i.e., the circuit is open).
Internal Resistance (\(r\))
Internal resistance (\(r\)) is the resistance within the power source itself (due to chemical resistance in a battery or components in a power supply).
When current (\(I\)) flows, some energy is inevitably wasted overcoming this internal resistance. This causes a "lost voltage" or lost potential difference (\(Ir\)).
Terminal Potential Difference (\(V\))
The voltage actually available to the external circuit is called the terminal potential difference (\(V\)).
The terminal PD is always less than the e.m.f. when current is flowing: $$\mathbf{V = \mathcal{E} - Ir}$$
The total resistance in the external circuit is \(R\). Since \(V=IR\), we can substitute this into the equation above: $$\mathbf{IR = \mathcal{E} - Ir}$$
Rearranging this gives the full circuit equation: $$\mathbf{\mathcal{E} = I(R+r)}$$
This equation shows that the total energy supplied by the source (\(\mathcal{E}\)) is used to overcome external resistance (\(IR\)) and internal resistance (\(Ir\)).
Key Takeaway: EMF vs. PD
EMF (\(\mathcal{E}\)) is the "potential when the switch is open" (total supply).
Terminal PD (\(V\)) is the "potential when the switch is closed" (voltage available externally).
The difference is the voltage "lost" inside the battery: \(\mathcal{E} - V = Ir\).