To solve this, let's go through the steps.

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Step 1: Draw the Parallel Circuit

A parallel circuit has each resistor connected directly to the voltage source, so each resistor has the same 15V across it. The circuit diagram would consist of a 15V battery and each resistor (R1, R2, R3, R4, and R5) connected in parallel.

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Step 2: Solve for the Equivalent Resistance ( $R_{eq}$ )

In a parallel circuit, the equivalent resistance $R_{eq}$ is found using the formula:

$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \frac{1}{R_4} + \frac{1}{R_5}$

Substituting the values:

$\frac{1}{R_{eq}} = \frac{1}{5} + \frac{1}{7} + \frac{1}{8} + \frac{1}{3} + \frac{1}{2}$

Calculating each term:

1. $\frac{1}{5} = 0.2$
2. $\frac{1}{7} \approx 0.1429$
3. $\frac{1}{8} = 0.125$
4. $\frac{1}{3} \approx 0.3333$
5. $\frac{1}{2} = 0.5$

Adding these up:

$\frac{1}{R_{eq}} \approx 0.2 + 0.1429 + 0.125 + 0.3333 + 0.5 = 1.3012$

Now, take the reciprocal to find $R_{eq}$:

$R_{eq} \approx \frac{1}{1.3012} \approx 0.768 \ \Omega$

So, the equivalent resistance is approximately 0.768 Ω.

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Step 3: Calculate the Total Current ( $I_{total}$ )

Using Ohm's law, $V = IR$, we find the total current by dividing the voltage by the equivalent resistance:

$I_{total} = \frac{V}{R_{eq}} = \frac{15}{0.768} \approx 19.53 \ \text{A}$

So, the total current flowing through the circuit is approximately 19.53 A.

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Step 4: Calculate the Current Through Each Resistor

Since each resistor is in parallel, each one has the full 15V across it. We use Ohm's law to find the current through each resistor:

1. Current through R1: $I_1 = \frac{V}{R_1} = \frac{15}{5} = 3 \ \text{A}$

2. Current through R2: $I_2 = \frac{V}{R_2} = \frac{15}{7} \approx 2.14 \ \text{A}$

3. Current through R3: $I_3 = \frac{V}{R_3} = \frac{15}{8} = 1.875 \ \text{A}$

4. Current through R4: $I_4 = \frac{V}{R_4} = \frac{15}{3} = 5 \ \text{A}$

5. Current through R5: $I_5 = \frac{V}{R_5} = \frac{15}{2} = 7.5 \ \text{A}$

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Summary of Results

• Equivalent Resistance ($R_{eq}$): 0.768 Ω
• Total Current ($I_{total}$): 19.53 A
• Current through Each Resistor:
  • $I_1 = 3 \ \text{A}$
  • $I_2 = 2.14 \ \text{A}$
  • $I_3 = 1.875 \ \text{A}$
  • $I_4 = 5 \ \text{A}$
  • $I_5 = 7.5 \ \text{A}$

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Would you like further details on any of these calculations?

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Follow-Up Questions

1. How does the total current change if another resistor is added in parallel?
2. What happens to the equivalent resistance if we remove one resistor from the circuit?
3. How would the power consumption of each resistor be calculated?
4. How does the equivalent resistance in a parallel circuit compare to the smallest resistor?
5. Can the total current ever be less than the current through any single resistor in a parallel circuit?

Tip

In parallel circuits, the equivalent resistance is always less than the smallest resistor in the circuit.