Series and parallel circuits Revision Notes

Electric current (I) is defined as the rate of flow of electric charge (Q) through a conductor or circuit component. It is measured in amperes (A), where 1 ampere equals 1 coulomb of charge passing a point per second.

Mathematically, current is expressed as: I = Q / t

where I is current in amperes (A), Q is charge in coulombs (C), and t is time in seconds (s).

• Electric current is a scalar quantity with units of amperes (A).
• It represents how much charge flows per unit time through a point in the circuit.

A series circuit is an electrical circuit in which components are connected end-to-end so that there is only one path for current to flow.

All components share the same current because the charge has only one route to travel.

If any component in the series circuit is disconnected or fails, the entire circuit is broken and current stops flowing.

• Components connected in a single loop form a series circuit.
• There are no branches or junctions where current can split.
• The current flows through each component sequentially.

Because there is only one path for the charge to flow, the current is the same at every point in a series circuit.

Charge cannot accumulate or be lost at any point; therefore, the rate of flow of charge (current) must be constant throughout.

This means the current leaving the power source equals the current through each component and the current returning to the source.

• Current is constant at all points in a series circuit.
• No branching means no division of current.
• Charge conservation ensures uniform current flow.

Since current is the same through all components in series, the total current is equal to the current through each individual component.

This allows for straightforward calculations of current in series circuits without needing to consider current division.

However, the potential difference (voltage) across each component can differ, depending on its resistance.

• Current I_total = I_component1 = I_component2 =...
• Voltage divides among components in proportion to their resistance.
• If one component changes resistance, current changes for the whole circuit.

To draw a series circuit showing uniform current: start by drawing a closed loop with a power supply (battery) symbol on one side.

Add components such as resistors or lamps connected one after another along the loop.

Label the current at several points along the loop with the same symbol (e.g., I) and indicate the direction of current flow with arrows following conventional current (positive to negative).

• Draw a single closed loop with no branches.
• Place components sequentially along the loop.
• Label current as constant (I) at multiple points.
• Use arrows to show direction of current flow from positive to negative terminal.

A series circuit is an electrical circuit in which components are connected end-to-end in a single path for the electric current to flow. The current passes through each component one after another.

A parallel circuit is an electrical circuit where components are connected across the same two points, creating multiple paths for the current to flow. Each component is connected directly to the voltage source.

Electric current (I) is the rate of flow of electric charge, measured in amperes (A). Potential difference (voltage, V) is the energy transferred per unit charge, measured in volts (V). Resistance (R) is the opposition to current flow, measured in ohms (Ω).

• Series circuit: components connected in a single path.
• Parallel circuit: components connected in multiple paths across the same voltage.

To build a series circuit, connect components such as resistors, lamps, or cells end-to-end so that there is only one continuous path for current. For example, connect the positive terminal of a battery to one terminal of a lamp, then connect the other terminal of the lamp to the next component, and so on, finally connecting back to the battery's negative terminal.

In a series circuit, the same current flows through all components because there is only one path.

When drawing a series circuit diagram, use standard symbols: a cell is represented by a long and a short parallel line, a resistor by a zigzag line, and wires by straight lines connecting components in a single loop.

• Connect components end-to-end forming one loop.
• Current is the same through all components.

To build a parallel circuit, connect components so that each one is connected directly across the voltage source, creating branches. For example, connect the positive terminal of a battery to a junction, then connect each component between this junction and a common return junction connected to the battery's negative terminal.

In a parallel circuit, the voltage across each component is the same because each branch is connected directly across the power supply.

When drawing a parallel circuit diagram, show branches splitting from a junction and rejoining at another junction, with each component on its own branch, using standard symbols for components and junctions represented by dots.

• Connect components across the same two points forming branches.
• Voltage across each component is the same.

In series circuits, the current (I) is the same at every point because there is only one path. The total potential difference (V_total) supplied by the source is shared among the components, so: V_total = V_1 + V_2 +... + V_n, where V_n is the voltage across each component.

In parallel circuits, the voltage across each branch is the same as the source voltage: V_total = V_1 = V_2 =... = V_n. The total current from the source (I_total) splits at the junctions and is the sum of the currents through each branch: I_total = I_1 + I_2 +... + I_n.

These behaviours are essential for analysing circuits and predicting how components will operate when connected in series or parallel.

• Series: current is the same; voltage divides among components.
• Parallel: voltage is the same across branches; current divides among branches.

In a series circuit, the total resistance (R_total) is the sum of the individual resistances: R_total = R_1 + R_2 +... + R_n.

This is because the current must pass through each resistor sequentially, increasing the overall opposition to current flow.

Example calculation: Two resistors of 4 Ω and 6 Ω are connected in series. Calculate the total resistance: R_total = 4 Ω + 6 Ω = 10 Ω. The unit is ohms (Ω).

• Total resistance in series is the sum of resistances.

To draw a series circuit diagram: start with a cell symbol, then connect components in a single loop using straight lines for wires. Place components one after another without branching.

To draw a parallel circuit diagram: start with a cell symbol, draw a junction splitting into branches, place one component on each branch, then reconnect branches at a second junction leading back to the cell.

Mark junctions with dots where wires split or join. Use standard symbols for cells, resistors, lamps, and wires. Label components and indicate polarity where relevant.

• Series: single loop, components in sequence.

Electromotive force (e.m.f.) is the energy supplied per unit charge by a source such as a battery or cell when no current is flowing. It is measured in volts (V). The e.m.f. represents the maximum potential difference the source can provide across its terminals.

Physically, e.m.f. arises from chemical or physical processes inside the source that move charges against an electric field, thus supplying energy to the charges.

• Unit of e.m.f.: volt (V)
• 1 volt = 1 joule per coulomb (1 V = 1 J/C)
• e.m.f. is the energy supplied per unit charge

When two or more sources of e.m.f. are connected end-to-end in a single loop (series), their voltages add up algebraically to give the total e.m.f. of the combination.

This is because the charges gain energy from each source sequentially as they move around the circuit, so the total energy supplied per coulomb is the sum of the energies supplied by each source.

The total e.m.f. depends on the polarity of each source. If all sources are connected with the same polarity (positive terminal to negative terminal), their e.m.f.s add directly. If any source is reversed, its e.m.f. subtracts from the total.

• Series connection: sources connected positive to negative in a chain
• Total e.m.f. is sum of individual e.m.f.s considering polarity
• Charges gain energy from each source sequentially

The total electromotive force (E_total) of n sources connected in series is given by:

\[ E_{total} = E_1 + E_2 + E_3 + \cdots + E_n \] where each \(E_i\) is the e.m.f. of the i-th source.

If a source is reversed, its e.m.f. is taken as negative in the sum.

• \( E_{total} = \sum_{i=1}^n E_i \)
• Consider polarity: reversed sources subtract from total e.m.f.
• Units: all e.m.f.s must be in volts (V)

Calculate the total e.m.f. when three cells of 1.5 V, 2.0 V, and 1.5 V are connected in series with the first two connected with the same polarity but the third reversed.

Step 1: Assign signs based on polarity: +1.5 V, +2.0 V, -1.5 V.

Step 2: Sum the e.m.f.s: \( E_{total} = 1.5 + 2.0 - 1.5 = 2.0 \, \text{V} \). The total e.m.f. is 2.0 V.

• Identify polarity and assign positive or negative signs
• Add e.m.f. values algebraically
• Result is total voltage available to the circuit

A frequent mistake is to add e.m.f.s without considering polarity, leading to incorrect total e.m.f. values.

Another error is mixing units or forgetting to convert millivolts to volts before summing.

Always check the orientation of each source and ensure consistent units before calculation.

• Check polarity of each source before adding
• Convert all e.m.f.s to volts if necessary
• Remember total e.m.f. affects the voltage supplied to the circuit

Resistors are said to be connected in series when they are connected end-to-end so that the same current flows through each resistor sequentially.

In a series circuit, there is only one path for the electric current to flow.

The total potential difference across the series combination is the sum of the potential differences across each resistor.

• Current is the same through all resistors in series.
• Potential difference divides among resistors in proportion to their resistance.
• Series connection means resistors are connected one after another in a single loop.

The total resistance, R_total, of resistors connected in series is the sum of their individual resistances.

This is because each resistor adds to the opposition to current flow, increasing the overall resistance.

The formula is: R_total = R_1 + R_2 + R_3 +... + R_n, where R_1, R_2, etc. are the resistances of each resistor.

• Total resistance increases as more resistors are added in series.
• Resistance is measured in ohms (Ω).
• The formula applies regardless of the number of resistors connected in series.

Resistance is measured in ohms, symbolised by Ω (capital omega).

The total resistance, R_total, and individual resistances, R_1, R_2, etc., are all expressed in ohms.

When performing calculations, ensure all resistance values are in the same unit (ohms) before adding.

• Resistance unit: ohm (Ω)
• Symbol for total resistance: R_total
• Symbols for individual resistors: R_1, R_2,..., R_n

Calculate the total resistance of three resistors connected in series with resistances 4.0 Ω, 6.0 Ω, and 10.0 Ω.

Step 1: Write down the formula: R_total = R_1 + R_2 + R_3

Step 2: Substitute the values: R_total = 4.0 Ω + 6.0 Ω + 10.0 Ω = 20.0 Ω (units included)

• Formula: R_total = R_1 + R_2 + R_3
• Substitution: R_total = 4.0 Ω + 6.0 Ω + 10.0 Ω
• Answer: R_total = 20.0 Ω

A common mistake is to multiply resistances instead of adding them in series.

Another error is to omit units or mix units (e.g., kilo-ohms and ohms) without conversion.

Always check that resistors are indeed in series before applying the formula, as parallel circuits require a different approach.

• Do not multiply resistances in series; add them.
• Convert all resistances to ohms before calculation.
• Confirm the circuit configuration is series before calculating total resistance.

Electric current is the rate of flow of electric charge through a conductor, measured in amperes (A). One ampere corresponds to one coulomb of charge passing a point per second.

Current is measured using an ammeter connected in series with the component or branch where the current is to be measured.

Current is a scalar quantity; it has magnitude but no direction, although conventional current flow is considered from positive to negative terminal of the power supply.

• Current (I) unit: ampere (A)
• 1 A = 1 coulomb/second
• Measured with an ammeter connected in series

A parallel circuit consists of two or more branches connected across the same two points, providing multiple paths for current to flow.

Each branch in a parallel circuit has the same potential difference (voltage) across it, equal to the voltage of the power supply.

Because branches are connected in parallel, the total current supplied by the source divides among the branches according to their resistances.

• Branches connected across same two points
• Voltage across each branch is equal
• Multiple current paths exist

In a parallel circuit, the total current from the power source is the sum of the currents flowing through each individual branch.

Mathematically, this is expressed as: I_total = I_1 + I_2 + I_3 +... where I_1, I_2, I_3 are currents in each branch.

Since current divides at junctions, the current in each branch is less than the total current supplied by the source, making the total current greater than any single branch current.

• Total current equals sum of branch currents: I_total = Σ I_branch
• Current divides at junctions
• Total current > current in any branch

The division of current occurs because electrons have multiple paths to flow through; the total charge flow per second is shared among branches.

If one branch is disconnected or breaks, current can still flow through the other branches, so the circuit does not stop functioning entirely.

The total resistance of a parallel circuit is less than the resistance of the smallest individual branch, allowing more total current to flow for a given voltage.

• Multiple paths allow current division
• Circuit remains functional if one branch breaks
• Total resistance in parallel is less than smallest branch resistance

Consider a parallel circuit with two branches: Branch 1 carries 2.0 A, Branch 2 carries 3.0 A.

To find the total current supplied by the source, add the currents in each branch: I_total = I_1 + I_2 = 2.0 A + 3.0 A = 5.0 A.

This shows the total current is greater than the current in either branch individually.

• I_total = I_1 + I_2
• Example: 2.0 A + 3.0 A = 5.0 A total current
• Total current > individual branch currents

Resistance is a measure of how much a component opposes the flow of electric current, measured in ohms (Ω). When resistors are connected in parallel, they provide multiple paths for current to flow. This arrangement affects the total resistance of the circuit differently compared to resistors connected in series.

In a parallel circuit, the voltage across each resistor is the same, but the current divides among the parallel branches. Because current has more than one path, the overall opposition to current flow (total resistance) decreases.

• Resistance (R) is measured in ohms (Ω).
• Voltage (potential difference) across parallel resistors is equal.
• Current divides among parallel branches.

When two resistors are connected in parallel, the current can flow through either resistor. This effectively increases the total cross-sectional area available for current flow, similar to how adding more lanes to a road allows more cars to pass simultaneously. As a result, the total resistance decreases because the circuit offers less opposition to the flow of charge.

The total resistance in parallel is always less than the resistance of the smallest resistor because the current has alternative routes, reducing the overall difficulty for current to flow.

• Multiple current paths reduce total resistance.
• Total resistance in parallel is less than the smallest individual resistor.

The total resistance (R_total) of two resistors connected in parallel, with resistances R₁ and R₂, is given by the formula:

\[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} \]

To find R_total, rearrange the formula: \[ R_{total} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}} \] or equivalently: \[ R_{total} = \frac{R_1 \times R_2}{R_1 + R_2} \]

• Formula for two resistors in parallel: \( R_{total} = \frac{R_1 R_2}{R_1 + R_2} \).
• Total resistance is always less than either R₁ or R₂.

Calculate the total resistance of two resistors connected in parallel: one resistor of 6 Ω and another of 3 Ω.

Step 1: Write down the formula: \[ R_{total} = \frac{R_1 R_2}{R_1 + R_2} \]

Step 2: Substitute values: \[ R_{total} = \frac{6 \times 3}{6 + 3} = \frac{18}{9} \] Ω.

• Step 3: Calculate: \[ R_{total} = 2 \, \Omega \].
• Answer: The total resistance is 2 Ω, which is less than both 3 Ω and 6 Ω.

A common mistake is to add resistances in parallel directly as if they were in series, which leads to an incorrect higher total resistance. Remember, in parallel, the total resistance decreases because the current has multiple paths.

Another misconception is that the total resistance can be greater than the largest resistor in parallel; this is never true. The total resistance is always less than the smallest resistor in the parallel network.

• Do not add resistors in parallel as \( R_1 + R_2 \).
• Total resistance in parallel is always less than the smallest resistor.

In a parallel circuit, components are connected across the same two points, creating multiple branches. Each branch provides an independent path for current to flow from the power supply.

When lamps are connected in parallel, each lamp is connected directly across the supply voltage. This means each lamp experiences the full potential difference of the power source, unlike in series circuits where voltage is divided among components.

• Multiple branches connected across the same voltage source.
• Each branch operates independently.
• Voltage across each branch is equal to the supply voltage.

The potential difference (voltage) across each lamp in a parallel circuit is the same as the supply voltage. For example, if the supply is 12 V, each lamp receives 12 V.

The total current from the power supply is the sum of the currents through each lamp branch. Each lamp draws current according to its resistance and the voltage across it, following Ohm's law: I = V/R.

Because voltage is constant across each lamp, the brightness of each lamp depends only on its resistance and remains consistent regardless of how many lamps are connected.

• Voltage across each lamp equals supply voltage (V).
• Total current equals sum of branch currents (I_total = I₁ + I₂ +...).
• Current through each lamp: I = V/R (Ohm's law).

In parallel circuits, each lamp operates independently. If one lamp fails (e.g., filament breaks), the other lamps continue to receive full voltage and remain lit.

This contrasts with series circuits, where failure of one lamp breaks the circuit and all lamps go out.

This independent operation improves reliability in lighting systems, ensuring continuous illumination even if one lamp fails.

• Failure of one lamp does not interrupt current in other branches.
• Each lamp branch is electrically separate from others.
• Improved reliability and safety in lighting applications.

Lamps can be added or removed from a parallel circuit without affecting the operation of other lamps.

Adding a lamp adds another branch, increasing total current but not changing voltage across existing lamps.

Removing a lamp simply removes that branch, leaving other lamps unaffected and maintaining their brightness.

• Adding lamps increases total current but not voltage per lamp.
• Removing lamps does not affect other lamps.
• Allows easy modification of lighting circuits.

Parallel connections are commonly used in household and commercial lighting because they ensure consistent brightness and reliability.

Each lamp receives the full supply voltage, so lamps designed for a specific voltage operate correctly.

The circuit design allows maintenance or replacement of lamps without turning off the entire system, improving convenience and safety.

• Common in domestic and commercial lighting installations.
• Ensures lamps operate at intended brightness.
• Facilitates maintenance without disrupting entire circuit.

In a parallel circuit, the electric current splits at junctions where the circuit branches. The principle of conservation of electric charge states that charge cannot accumulate at a junction. Therefore, the total current entering a junction must equal the total current leaving it.

Mathematically, if I_total is the current entering the junction, and I_1, I_2,..., I_n are the currents leaving through each branch, then:

I_total = I_1 + I_2 +... + I_n. This is known as Kirchhoff's Current Law (KCL).

• Current (I) is the rate of flow of electric charge, measured in amperes (A).
• At any junction, total incoming current equals total outgoing current.
• This principle ensures charge conservation in circuits.

In a series circuit, components are connected end-to-end so that there is only one path for current flow. The total potential difference (p.d.) supplied by the source is shared among the components.

The total potential difference across the series components equals the sum of the individual potential differences across each component:

V_total = V_1 + V_2 +... + V_n, where V is measured in volts (V). This follows from the conservation of energy: the energy supplied per coulomb by the source is used up across the components in series.

• Potential difference (p.d.) is the energy transferred per unit charge, measured in volts (V).
• In series, p.d. divides among components according to their resistance.
• Sum of p.d.s across components equals total p.d. supplied.

In a parallel circuit, components are connected across the same two points, creating multiple branches. Each branch is connected directly to the voltage source.

The potential difference across each parallel branch is the same and equal to the potential difference across the source:

V_total = V_branch1 = V_branch2 =... = V_branchn. This is because each branch is connected directly across the source terminals.

• Each parallel branch has the same p.d. as the source.
• Current divides among branches, but p.d. remains constant across them.
• This principle is essential for analysing parallel circuits.

A parallel circuit has a total current of 6.0 A entering a junction. Two branches leave the junction with currents of 2.5 A and 1.5 A respectively. Calculate the current in the third branch.

Using conservation of current: I_total = I_1 + I_2 + I_3

Rearranged: I_3 = I_total - I_1 - I_2 = 6.0 A - 2.5 A - 1.5 A = 2.0 A.

• Formula: I_total = sum of branch currents.
• Rearrangement to find unknown current.
• Substitution with units.
• Answer: 2.0 A.

Three resistors are connected in series across a 12 V battery. The potential differences across the first two resistors are 3.5 V and 4.0 V respectively. Calculate the potential difference across the third resistor.

Total p.d. supplied: V_total = 12 V. Sum of known p.d.s: 3.5 V + 4.0 V = 7.5 V.

Potential difference across third resistor: V_3 = V_total - (V_1 + V_2) = 12 V - 7.5 V = 4.5 V.

• Formula: V_total = V_1 + V_2 + V_3.
• Rearranged to find V_3.
• Substitution with units.
• Answer: 4.5 V.

Two resistors are connected in parallel across a 9 V battery. Calculate the potential difference across each resistor.

Since resistors are in parallel, the potential difference across each resistor equals the battery voltage:

V_1 = V_2 = 9 V. No calculation is needed beyond recognising the principle.

• Potential difference is the same across all parallel branches.
• Each resistor has p.d. equal to the source voltage.
• Answer: 9 V across each resistor.

Electric current (I) is defined as the rate of flow of electric charge (Q) through a conductor, measured in amperes (A). One ampere corresponds to one coulomb of charge passing a point per second (1 A = 1 C/s).

Electric charge is conserved, meaning it cannot be created or destroyed in an isolated system. In an electric circuit, this conservation implies that the amount of charge flowing into any point must equal the amount flowing out, ensuring continuous current flow without accumulation of charge.

• Current (I) = charge flow per unit time (Q/t)
• Unit of current: ampere (A)
• Conservation of charge: charge is neither created nor destroyed

A junction in an electric circuit is a point where three or more conductors meet, allowing current to split or combine.

The fundamental rule is: the total current entering a junction equals the total current leaving the junction. Mathematically, if currents I₁, I₂,..., Iₙ enter and currents Iₐ, I_b,..., I_m leave, then:

\[ \sum_{in} I = \sum_{out} I \] This ensures no charge accumulates at the junction, consistent with charge conservation.

• Junction: point where conductors meet
• Total current in = total current out
• No charge accumulation at junction

In parallel circuits, the current from the power source divides among the different branches at junctions.

For example, if a current I enters a junction and splits into two branches with currents I₁ and I₂, then:

\[ I = I_1 + I_2 \] This can be extended to any number of branches: the sum of branch currents equals the incoming current at the junction.

• Current divides at junctions in parallel circuits
• Sum of branch currents equals total current entering junction
• Useful for calculating unknown branch currents

To draw a circuit junction illustrating current conservation:

1. Draw a point representing the junction on paper.

2. Draw three or more wires meeting at this point, each labelled with current arrows showing direction (towards or away from the junction). For example, one wire with current I entering the junction, two wires with currents I₁ and I₂ leaving the junction.

• Draw junction point
• Draw wires meeting at junction
• Label currents with arrows indicating direction

A current of 5.0 A enters a junction and splits into two branches. One branch carries 3.0 A. Find the current in the other branch.

Using conservation of current: total current in = total current out

\[ 5.0\,A = 3.0\,A + I_2 \] Rearranging: \[ I_2 = 5.0\,A - 3.0\,A = 2.0\,A \] The current in the other branch is 2.0 A.

• Use total current in = total current out
• Rearrange to find unknown current
• Substitute values and calculate

When resistors are connected in parallel, they are connected across the same two points, providing multiple paths for electric current to flow. Each resistor branch experiences the same potential difference (voltage) as the source.

Because current can split and flow through each resistor separately, the total current supplied by the source increases compared to a single resistor.

The total resistance of the parallel combination is always less than the resistance of the smallest individual resistor in the circuit.

• Parallel connection means resistors share the same voltage.
• Current divides among parallel branches.
• Total resistance decreases with more parallel resistors.

The total resistance, R_total, of two resistors connected in parallel with resistances R₁ and R₂ is given by the reciprocal formula:

\[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} \]

This formula can be extended for more than two resistors, but here we focus on two resistors only.

• Use reciprocal sums to find total resistance.
• Total resistance is less than each individual resistor.

Suppose two resistors, R₁ = 6 Ω and R₂ = 3 Ω, are connected in parallel. Calculate the total resistance:

1. Write the formula: \( \frac{1}{R_{total}} = \frac{1}{6} + \frac{1}{3} \)

2. Calculate each term: \( \frac{1}{6} = 0.1667 \), \( \frac{1}{3} = 0.3333 \) (units: Ω⁻¹) 3. Add the reciprocals: \( 0.1667 + 0.3333 = 0.5 \) 4. Take the reciprocal of the sum to find total resistance: \( R_{total} = \frac{1}{0.5} = 2 \, \Omega \) 5. State the answer with units: Total resistance is 2 Ω.

• Show formula substitution with units.
• Calculate reciprocals carefully.
• Add reciprocals and invert to find total resistance.
• Answer must include correct units (ohms, Ω).

Because the total resistance is less than the smallest resistor, the circuit allows more current to flow than through either resistor alone.

This is because the current has multiple paths, reducing the overall opposition to current flow.

The voltage across each resistor in parallel is the same as the source voltage, which differs from series circuits where voltage divides among components.

• Total resistance decreases with parallel resistors.
• Voltage across each resistor is equal in parallel.
• Current divides among parallel branches.

A frequent mistake is to add resistances directly as in series circuits, which is incorrect for parallel connections.

Another error is forgetting to take the reciprocal of the sum of reciprocals to find total resistance.

Always include units in calculations to avoid confusion and loss of marks.

• Do not add resistors directly in parallel.
• Remember to invert the sum of reciprocals.
• Include units in every step of calculation.