Math Problem Statement

In a series circuit with resistances of 8 52, 12 S, and 20 S2, if the circuit is powered by a 40 V supply, what is the current in the circuit and the total power dissipated?

Solution

In a series circuit, the total resistance is the sum of the individual resistances, and the current can be calculated using Ohm's Law. The power dissipated in the circuit can be found using the formula for power.

Step 1: Find the total resistance

The total resistance RtotalR_{total} in a series circuit is the sum of the individual resistances:

Rtotal=R1+R2+R3R_{total} = R_1 + R_2 + R_3

Given:

  • R1=8ΩR_1 = 8 \, \Omega
  • R2=12ΩR_2 = 12 \, \Omega
  • R3=20ΩR_3 = 20 \, \Omega

Rtotal=8Ω+12Ω+20Ω=40ΩR_{total} = 8 \, \Omega + 12 \, \Omega + 20 \, \Omega = 40 \, \Omega

Step 2: Find the current using Ohm's Law

Ohm's Law states that the current II is given by:

I=VRtotalI = \frac{V}{R_{total}}

Where:

  • V=40VV = 40 \, \text{V} (the supply voltage)
  • Rtotal=40ΩR_{total} = 40 \, \Omega

I=40V40Ω=1AI = \frac{40 \, \text{V}}{40 \, \Omega} = 1 \, \text{A}

So, the current in the circuit is 1A1 \, \text{A}.

Step 3: Find the total power dissipated

The power dissipated in a circuit can be calculated using the formula:

P=V×IP = V \times I

Where:

  • V=40VV = 40 \, \text{V}
  • I=1AI = 1 \, \text{A}

P=40V×1A=40WP = 40 \, \text{V} \times 1 \, \text{A} = 40 \, \text{W}

Final Answers:

  • The current in the circuit is 1A1 \, \text{A}.
  • The total power dissipated is 40W40 \, \text{W}.

Would you like more details on any of the steps or have any questions?


Here are 5 related questions you might find interesting:

  1. How would the current change if the resistance values were doubled?
  2. What happens to the total power dissipated if the voltage supply is increased to 80V?
  3. How would the current and power be affected if the resistors were connected in parallel instead of series?
  4. How do you calculate the voltage drop across each resistor in a series circuit?
  5. What is the difference between power dissipated and power supplied in a circuit?

Tip: Always check the configuration of the resistors (series or parallel) before applying formulas, as the calculations for total resistance are different in each case.

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Math Problem Analysis

Mathematical Concepts

Ohm's Law
Power Calculation
Series Circuits

Formulas

R_total = R_1 + R_2 + R_3
I = V / R_total
P = V * I

Theorems

Ohm's Law

Suitable Grade Level

Grades 9-11