Question

The power in a resistor of resistance \(R\) in which a current \(i\) flows is \(i^{2} R .\) If the current is reduced to one-third its former value, the power will be $$\left(\frac{i}{3}\right)^{2} R$$ Simplify this expression.

Short answer

The simplified expression for the new power when the current is reduced to one-third is P' = (i^2/9) R.

Step-by-step solution

Substitute the reduced current into the power formula

The original power expression is given by the formula P = i^2 R. To find the new power when the current is reduced to a third, substitute i/3 for i in the power formula. This gives the new power expression as P' = (i/3)^2 R.

Expand the power of the current

Expand the term (i/3)^2 by squaring both the numerator and the denominator. This gives (i^2/9).

Combine the squared current with the resistance

Multiply the expanded current term by the resistance R to get the new power expression: P' = (i^2/9) R.

Simplify the expression

Since there are no like terms to combine or further simplification to do, the simplified expression for the new power is P' = (i^2/9) R.

Key concepts

Understanding Electrical Resistance

Electrical resistance is a fundamental concept in the study of electricity. It's a property that quantifies how much an object opposes the flow of electric current. Think of it as a hurdle that electrons have to jump over as they move through a conductor. The higher the resistance, the harder it is for the current to flow, and subsequently the more heat is generated.

Resistance, represented with the symbol 'R', is measured in ohms (Ω). When the current 'i' flows across a resistor with resistance 'R', the power exerted, given by the formula 'P = i^2 R', can tell us how much energy per second is being converted into heat or another form of energy within that resistor.

Understanding the relationship between resistance and current is key to mastering many concepts in electrical engineering as well as in physics. It explains, for example, why high-powered appliances require thicker wires – to reduce resistance and therefore the heat produced.

The Flow of Electric Current

Electric current represents the flow of electric charge, and it's similar to the flow of water through a pipe. Just as the amount of water flowing through a pipe could change, electric current can vary in size as well. Measured in amperes (A), current is a vital part of the equation when calculating power in electrical circuits.

The value of the current 'i' flowing through a resistor directly affects the power dissipated by that resistor. Mathematically, we express the power 'P' as the product of the square of the current and the resistance (P = i^2 R). Thus, if you decrease the current, the power will also decrease. As shown in the exercise, reducing the current to one-third affects the power dramatically, bringing it down to just one-ninth of its original value because of the squaring of the fractional current.

Power Simplification in Resistive Circuits

Power simplification in electrical circuits often involves manipulating equations to find a more straightforward or insightful form. A common task is to understand how changes in current or resistance impact power, P, as shown in the exercise.

When we reduce the current to one-third, the formula for power becomes \(\left(\frac{i}{3}\right)^2 R\). Simplifying the expression by squaring the fraction means multiplying the current squared by the squared denominator: \(\frac{i^2}{9}\). The resistor's resistance 'R' doesn't change, so the new power is expressed as \(P' = \frac{i^2 R}{9}\). This simplification shows that the resistor's power is directly proportional to the square of the current, illustrating why even small changes in current can result in significant changes in power dissipated by a resistor.

Remember that the squared relationship between current and power is why electrical systems must be carefully designed to manage current flow and avoid excessive power dissipation, which can lead to overheating and potential damage.