Problem 52 - Algebra A Combined Function

Question

Solve each equation for the indicated variable. \(W=\frac{C E^{2}}{2}\) for \(C\) (Electronics: energy stored in a capacitor)

Step-by-Step Solution

Answer

The solution for \( C \) is \( C = \frac{2W}{E^2} \).

Step 1: Identify the Formula and Target Variable

The given equation is \( W = \frac{CE^2}{2} \). Our goal is to solve for the variable \( C \).

Step 2: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by 2 to get: \( 2W = CE^2 \).

Step 3: Isolate the Target Variable

Divide both sides of the equation by \( E^2 \) to isolate \( C \): \( C = \frac{2W}{E^2} \).

Key Concepts

Isolating Variables

Isolating a variable means rearranging an equation to make the variable the subject. In simpler terms, you want to "get" the variable on one side of the equation while everything else moves to the other side.

The key to isolating variables is to perform operations that do not change the equality of the equation. You might add, subtract, multiply, or divide both sides by the same number or expression.

Think of it as a balancing act: whatever you do to one side, you must also do to the other.

• Start by identifying the target variable you need to isolate;
• Move terms involving the variable to one side of the equation;
• Perform the necessary operations to reveal the target variable alone.

This method is fundamental in algebra and helps simplify solving equations.

Fraction Elimination

Fractions can make equations look complex, but don't worry! They can be eliminated by finding a common denominator or through equivalent operations. By clearing fractions, you make the equation simpler and easier to manipulate.

In our example equation, we have a fraction where the variable is part of the numerator. The fraction is \( \frac{CE^2}{2} \) and is set equal to \( W \).

To eliminate the fraction, multiply both sides by the denominator (in this case, 2). This step effectively removes the fraction:

• Resulting in \( 2W = CE^2 \)

This action clears the fraction, making it easier to proceed to isolate the variable.

Energy in a Capacitor

The formula \( W = \frac{C E^2}{2} \) describes energy stored in a capacitor. Capacitors store electrical energy in an electric field, and this energy can be expressed using this equation.

Here, \( W \) represents the energy (in joules), \( C \) is the capacitance (in farads), and \( E \) is the voltage (in volts).

Capacitance measures a capacitor's ability to store charge at a given voltage. The equation highlights how energy relates to both the capactiance and the square of the voltage.

• Higher capacitance or voltage results in more energy stored.
• The square of voltage emphasizes voltage's strong impact on energy.

Understanding this equation is crucial in electronics to determine how a capacitor will perform in a circuit.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations. These skills are essential for solving equations like the one provided.

When manipulating an equation, follow these steps:

• Perform the same operation on both sides to maintain equality.
• Use inverse operations to isolate variables.
• Simplify expressions as much as possible at each step to make calculations manageable.

In our example, first multiply by 2 to eliminate the fraction, and then divide by \( E^2 \) to isolate the variable \( C \). The resulting equation, \( C = \frac{2W}{E^2} \), is a clear expression for \( C \) in terms of known quantities.