Resistor Power Calculations

1. The Three Forms of the Power Equation

In our first lesson, we defined electrical power as the product of voltage and current.

$P=V\times I$

Where:

• $P$ is power in Watts (W)
• $V$ is voltage in Volts (V)
• $I$ is current in Amperes (A)

This is the foundational definition. Now, by using the algebraic power of Ohm's Law ($V=IR$), we can derive two other extremely useful forms of this equation. This process of substitution is a common technique in circuit analysis.

To see how this is done, let's turn to a clear textual explanation.

Let's summarize that key derivation:

1. To get Power from Current and Resistance:
   We start with $P=VI$. From Ohm's Law, we know that $V=IR$. If we substitute IR for V in the power equation, we get:
   $P=(I\cdot R)\cdot I$
   $\to P=I^{2}R$

2. To get Power from Voltage and Resistance:
   Again, we start with $P=VI$. This time, we rearrange Ohm's Law to solve for current: $I=V/R$. Now we substitute V/R for I in the power equation:
   $P=V\cdot \left(\frac{V}{R}\right)$
   $\to P=\frac{V^2}{R}$

So, we now have three equivalent tools at our disposal:

The formula you choose simply depends on which two quantities you already know.

2. Resolving a Common Confusion

Having three formulas can sometimes lead to a bit of confusion. Look at the last two:

• $P=I^{2}R$ seems to imply that power increases as resistance increases.
• $P=V^2/R$ seems to imply that power decreases as resistance increases.

So, which is it? Does increasing a resistor's value generate more or less heat? The answer, beautifully, is "it depends on what you hold constant."

This is a subtle but critical point for developing a deep intuition about circuit behavior. Let's watch a video that explores this question by considering two distinct physical scenarios.

As the video explained:

• If voltage is held constant (e.g., you connect a different resistor to the same battery), increasing the resistance will decrease the current (Ohm's Law). The drop in current is more significant than the increase in resistance, so the total power decreases. This scenario is described by $P=V^2/R$.
• If current is held constant (e.g., you have a special power supply that adjusts its voltage to maintain a constant current), increasing the resistance will require a larger voltage drop across the resistor. Both $R$ and $V$ increase, so the power increases. This scenario is described by $P=I^{2}R$.

In most simple DC circuits we'll be looking at, the voltage source is constant (like a battery), so the first scenario is more common.

3. Putting it into Practice: Calculation Examples

Now that we have the theory and intuition, let's use the formulas to solve some problems.

Let's consider a simple circuit: a 9V battery is connected to a 250$\Omega$ resistor.

Our goal is to find the power dissipated by the resistor.

Step 1: Identify your knowns.
We know $V=9\text{V}$ and $R=250\Omega$.

Step 2: Choose the best formula.
Since we know voltage and resistance, the most direct formula to use is $P=V^2/R$.

Step 3: Calculate.
$P=\frac{V^2}{R}=\frac{(9\text{V}{)}^2}{250\Omega }=\frac{81}{250}=0.324\text{W}$

Verification: What if we wanted to use one of the other formulas? We could! We would just need to find the current first using Ohm's Law.
$I=\frac{V}{R}=\frac{9\text{V}}{250\Omega }=0.036\text{A}$
Now we can use the other two power formulas to check our work:

• Using $P=VI$:
  $P=(9\text{V})\cdot (0.036\text{A})=0.324\text{W}$
• Using $P=I^{2}R$:
  $P=(0.036\text{A}{)}^2\cdot (250\Omega )=0.001296\cdot 250=0.324\text{W}$
  All three formulas give the same answer, as expected.

For more practice, watch this short segment of a problem-solving video.

Conclusion

In this lesson, we combined Ohm's Law with the definition of power to create a powerful and flexible set of analytical tools.

Key Takeaways:

• Power dissipated by a resistor can be calculated using three interchangeable formulas: $P=VI$, $P=I^{2}R$, and $P=V^2/R$.
• The latter two formulas are derived from $P=VI$ by substituting for voltage or current using Ohm's Law.
• The choice of which formula to use is a matter of convenience, based on the known quantities in the circuit.
• This dissipated power manifests primarily as heat. For this reason, physical resistors have power ratings (e.g., 1/4 W, 1/2 W, 1 W) which must not be exceeded, or the component will be destroyed.

So far, we've only dealt with very simple circuits containing a single source and a single resistor. Real-world circuits are rarely so simple. They have multiple components and multiple paths for current to flow. In our next lesson, we will introduce our first tool for analyzing these more complex circuits: Kirchhoff's Current Law (KCL). This law governs how current behaves at the junctions where components meet.