How much power is the 20ω resistor dissipating?

The 10Ω resistor in the figure (Figure 1) is dissipating 20 W of power.
How much power is the 5Ω resistor dissipating?
How much power is the 20Ω resistor dissipating?
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Answer

General guidance

Concepts and reason
The concepts used to solve the given problem are, equivalent resistance for series connection, Ohm’s law, and power dissipated through a resistor.

Use the expression for the power dissipated through a resistor and the property of current in series combination to calculate the power dissipated through the resistor. Next, use the expression for power dissipated through resistor and equivalent resistance in series connection, and Ohm’s law to calculate the power dissipated through the resistor.

Fundamentals

Write the expression for the power dissipated through a resistor.

P=1°R

Here, is the resistance of the resistor, is the power dissipated through the resistor, and is the current flowing through the resistor.

Write the expression for the equivalent resistance in series combination.

R., = R + R, +...+R,

Here, R,
RR are the series-connected resistances.

In series connection, the current is same through the circuit.

In parallel connection, the voltage is same in the circuit.

Ohm’s law states that the current through a resistor varies directly with the applied voltage across the end terminals of the resistor.

Write the expression for Ohm’s law.

I c V
V = IR

Here, is the resistance, is the voltage, and is the current through the resistor.

Step-by-step

Step 1 of 2

Draw the circuit diagram for the given problem.

mm
2012
100

Here, a, b and c are the resistors with resistances of , and , respectively. , , and are the current flowing through resistors a, b and c, respectively.

From the diagram given in the question, and are connected in series. Thus,

I, = 12

Write the expression for the power dissipated through resistor b.

P2 = 1;R,

Here, is the resistance of resistor b and is the power dissipated through resistor b.

Substitute for and for .

20 W=1} (102)

Rearrange for .

20 W
1,
10Ω
=1.414Α

Write the expression for the power dissipated through resistor a.

P=1R

Here, is the power dissipated through resistor a, and is the resistance of resistor a.

Substitute for .

P=IR,

Substitute 1.414 A for and for .

P. = (1.414 A) 512
= 10 W

The power dissipated through resistor is .


The power through resistor b is given, so use this value to calculate the current through resistor b. The resistors, a and b, are connected in series, so current flowing through both the resistors must be the same. Therefore, the current is already calculated for resistor a. Use the expression for power dissipated to calculate the power dissipated through resistor a.

Step 2 of 2

Redraw the circuit diagram for the given problem.

mm
2012
100

From the above diagram, resistors a and b are connected with the resistor c in parallel combination. Thus,

V. =V2

Here, is the voltage across the left segment of the circuit or the voltage through the series connection of resistors a and b; and is the voltage across the right segment of the circuit or the voltage across resistor c.

The resistors in the left segment of the circuit are in series combination. Thus, the equivalent resistance of the left segment is,

R, = R, +R,

Substitute for and for .

R, = 5Ω+10Ω
= 15 Ω

From the diagram, current through the left segment of the circuit is,

1= 1, or, 1 =12

Write the expression for the voltage in the left segment of the circuit in terms of resistance.

(u)1 =!1

Here, is the current through the left segment of the circuit.

Substitute for .

V =1,(R)

Substitute for and 1.414 A for .

V, = (1.414 A)(1512)
= 21.21 V

From Ohm’s law for resistor c.

Here, is the resistance of resistor c.

Write the expression for the power dissipated through resistor c.

P, = IR,

Here, is the power dissipated through resistor c.

Substitute for .

행화

Substitute for .

Substitute 21.21 v for and for .

B - (21.21 v)?
20 Ω
= 22.49 W
22.5 W

The power dissipated through resistor is 22.5 W.


The voltage across the circuit is the same for parallel connection; therefore, voltage is same for the right and left segments of the circuit. So, calculate the voltage across the left segment of the circuit and substitute the same value for the voltage across the resistor, c, because it is the only resistor in the right segment of the circuit.

The value of the current through resistor c is not known in the problem, and thus, from Ohm’s law, use appropriate conversion and substitute the voltage in the expression for power dissipated to calculate the power dissipated through resistor c.

Answer

The power dissipated through resistor is .

The power dissipated through resistor is 22.5 W.

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