A circuit has a resistance of 12 ohms and an impedance of 15 ohms. The power factor of the circuit w

Q: What exactly is the power factor?

The power factor is the cosine of the phase angle between the voltage and current in an AC circuit. It represents the ratio of real power (used to do work) to apparent power (total power supplied).

Q: Why do we use the formula $\frac{R}{Z}$?

In an impedance triangle, the resistance $R$ forms the base (adjacent side to the phase angle $\phi$), and the total impedance $Z$ forms the hypotenuse. Since $\cos\phi = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, the power factor is $\frac{R}{Z}$.

Q: Can the power factor ever be greater than 1?

No. Since the impedance $Z = \sqrt{R^2 + X^2}$, the resistance $R$ can never be greater than $Z$. Therefore, the ratio $\frac{R}{Z}$ will always be less than or equal to $1$.

Q: What does a power factor of 0.8 physically mean?

It means that $80\%$ of the total power supplied to the circuit is being converted into useful work (real power), while the remaining $20\%$ is reactive power that oscillates back and forth without doing work.

Q: What would the power factor be if the circuit had no reactance?

If there is no reactance ($X = 0$), then the impedance $Z$ equals the resistance $R$. The power factor would be $\frac{R}{R} = 1$, which is the ideal scenario for power transmission.

Question

A circuit has a resistance of $12 \Omega$ and an impedance of $15 \Omega$. The power factor of the circuit will be

Options:

(A) 0.8

(B) 0.4

(C) 1.25

(D) 0.125

Sigma Percentile · Accepted Answer

### Analyzing the Setup
Imagine you are looking at the core characteristics of an alternating current circuit. We are given two fundamental properties: the resistance $R$ and the total impedance $Z$. The resistance is the opposition to current flow that dissipates energy as heat, given as $12 \, \Omega$. The impedance, on the other hand, is the total opposition to the current, combining both resistance and reactance, given as $15 \, \Omega$.

Our goal is to find the **power factor** of this circuit. The power factor is a crucial metric in AC circuits because it tells us how effectively the circuit converts electrical power into useful work.

### The Master Equation
To find the power factor, we can visualize the circuit's properties using an **impedance triangle**. In this right-angled triangle, the base represents the resistance $R$, the perpendicular represents the net reactance $X$, and the hypotenuse represents the total impedance $Z$.

The angle between the resistance and the impedance is the phase angle $\phi$. The power factor is mathematically defined as the cosine of this phase angle. From our triangle, using basic trigonometry, the cosine of $\phi$ is the ratio of the adjacent side to the hypotenuse.

Therefore, our master equation is:
$$ 	ext{Power Factor} = \cos\phi = \frac{R}{Z} $$

### Final Calculation
Now, we simply substitute the given values into our master equation. We know $R = 12 \, \Omega$ and $Z = 15 \, \Omega$.

$$ \cos\phi = \frac{12}{15} $$

Both the numerator and the denominator are divisible by $3$. Let's simplify the fraction:

$$ \cos\phi = \frac{4}{5} $$

Finally, converting this fraction into a decimal gives us our answer:

$$ \cos\phi = 0.8 $$

The power factor of the circuit is **$0.8$**. This indicates that $80\%$ of the apparent power supplied to the circuit is being effectively used as real power.

Animated Solution for Physics - Electromagnetic Induction: A circuit has a resistance of $12Ω$ and an impedance of $15Ω$ . The power factor of the circuit will be

Select Answer:

Visualized Solution

$Impedance Triangle$

$Power Factor Formula$

$Substituting Values$

$Calculation$

$Conclusion$

$The Way Forward$

The Sigma Insight: Alternating Current (AC) and Voltage

Analyzing the Setup Imagine you are looking at the core characteristics of an alternating current circuit

The Master Equation To find the power factor, we can visualize the circuit's properties using an impedance triangle

Final Calculation Now, we simply substitute the given values into our master equation

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Animated Solution for Physics - Electromagnetic Induction: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A circuit has a resistance of 12Ω and an impedance of 15Ω. The power factor of the circuit will be

Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Impedance Triangle

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Power Factor Formula

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Substituting Values

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Way Forward