Question

When a current \(I\) causes a power \(P\) in a resistance, \(R,\) that resistance is given by $$ R=\frac{P}{I^{2}} $$ Write this equation without fractions.

Short answer

\(R I^2 = P\)

Step-by-step solution

Multiply both sides by the denominator

To remove the fraction, multiply both sides of the equation by the denominator, which is \(I^2\). This gives you \(I^2 \cdot R = P\).

Express the equation as a product

Rewrite the left-hand side of the equation as a single product: \(R \cdot I^2 = P\).

Key concepts

Understanding Ohm's Law

Ohm's Law is one of the foundational principles in electrical engineering and electronics. It defines the relationship between voltage (V), current (I), and resistance (R) in an electrical circuit. Ohm's Law is mathematically expressed by the equation:
\( V = I \times R \).
This simple formula explains how, for a given resistance, the electric current flowing through a circuit is directly proportional to the voltage across it. What makes Ohm's Law particularly useful is its versatility in solving for any one of the three variables if the other two are known.
For instance, if you need to calculate the resistance in a circuit, and you know the voltage and current, you could rearrange the formula to solve for resistance:
\( R = \frac{V}{I} \).
Ohm's Law is crucial for not just calculating resistance but also for understanding how electrical components will behave under different currents and voltages. When using Ohm's Law, remember to maintain consistent units (volts for voltage, ohms for resistance, and amperes for current) and use precision measuring tools to ensure accuracy in your calculations.

Applying the Power Formula

The Power formula is another vital concept in electrical studies, interrelating power (P), voltage (V), current (I), and resistance (R). In its most common form, the formula looks like this:
\( P = V \times I \).
This relation is a direct statement of how power, measured in watts (W), is the product of voltage, in volts (V), and current, in amperes (A). However, we can use Ohm's Law to express power in terms of resistance and current:
\( P = I^2 \times R \),
which is derived by substituting the voltage in the original power equation with Ohm’s law (\( V = I \times R \).) By doing this, we can understand how an increase in either current or resistance will result in a higher power dissipation in the circuit, which is useful when designing systems to avoid overheating or when troubleshooting electrical issues.
For students working with circuit design or analysis, mastering the Power formula and its applications is key to predicting and managing the behavior of electronic systems.

Algebraic Manipulation in Electrical Calculations

Algebraic manipulation is an essential tool in mathematics, particularly in solving physics and engineering problems such as electrical resistance calculations. Algebraic techniques allow students to rearrange equations and solve for unknown variables, making it possible to deduce a wide array of information from a limited set of data.
For example, the algebraic manipulation required to rewrite the resistance calculation formula from its fraction form:

Original Form

\( R = \frac{P}{I^{2}} \).

Manipulated Form

\( R \times I^2 = P \).
By multiplying both sides of the equation by the denominator, in this case, the square of the current (\( I^2 \)), we can remove the fraction and express resistance as a product of itself and current squared, directly related to power. This simple, yet powerful, algebraic operation transforms complex equations into more straightforward expressions that can be easily interpreted and solved.
Understanding and effectively applying algebraic manipulation is thus not only central to mathematical problem-solving but also to the practical application of such solutions in the real world of electrical engineering and circuitry.