Problem 54 - Physics

Question

A wire whose resistance is \(R\) is cut into three equally long pieces, which are then connected in parallel. In terms of \(R\), what is the resistance of the parallel combination?

Step-by-Step Solution

Answer

The resistance of the parallel combination is \( \frac{R}{9} \).

Step 1: Determine the resistance of each piece

Since the wire is cut into three equal pieces, the resistance of each piece, in terms of the original resistance \( R \), will be \( \frac{R}{3} \). This is because resistance is directly proportional to the length of the wire, so dividing the wire into three equal parts means that each part will have a resistance of one-third of the original.

Step 2: Apply the formula for parallel resistors

The total resistance \( R_t \) of resistors in parallel is given by the reciprocal of the sum of the reciprocals of their resistances: \[\frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\]Substitute \( R_1 = R_2 = R_3 = \frac{R}{3} \) into the equation:\[\frac{1}{R_t} = \frac{1}{\frac{R}{3}} + \frac{1}{\frac{R}{3}} + \frac{1}{\frac{R}{3}}\]

Step 3: Simplify the equation

Now simplify the equation:\[\frac{1}{R_t} = 3 \left(\frac{1}{\frac{R}{3}}\right) = 3 \times \frac{3}{R} = \frac{9}{R}\]

Step 4: Find the parallel resistance

Take the reciprocal to find \( R_t \):\[R_t = \frac{R}{9} \]Thus, the resistance of the parallel combination is \( \frac{R}{9} \).

Key Concepts

Resistance Calculation

To understand how cutting a wire affects its resistance, it's essential to know that resistance depends on both material and geometry. When a wire with resistance \( R \) is cut into smaller pieces, the resistance of each piece also changes. Since resistance is directly proportional to the length, if the wire is divided into three equal parts, the length of each part is one-third of the original. This means each part has a resistance of \( \frac{R}{3} \). By calculating it this way, you can see the relationship between the full wire resistance and the resistance of its segments. It's a basic principle but crucial for understanding how parallel circuits are constructed.

Parallel Resistors

When resistors are connected in parallel, the total or effective resistance decreases. The formula used to calculate total resistance in a parallel circuit is the reciprocal of the sum of the reciprocals of the individual resistances. The equation is: \[\frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\] In our scenario where each segment of the wire has a resistance \( \frac{R}{3} \), substituting these values into the equation gives you: \[\frac{1}{R_t} = \frac{1}{\frac{R}{3}} + \frac{1}{\frac{R}{3}} + \frac{1}{\frac{R}{3}}\]Simplifying this leads to the total parallel resistance being a fraction of the original total. This illustrates that the more parallel paths there are, the easier it is for electricity to flow, resulting in a smaller total resistance.

Cutting Wire into Parts

Cutting a wire into equal parts transforms the original resistor into multiple equivalent resistors. In practical applications, the length of the wire directly influences its resistance. By cutting a wire of resistance \( R \) into three parts, the individual segments each become smaller resistors with resistance \( \frac{R}{3} \).This setup is particularly beneficial when designing circuits that require lower total resistance or specific electrical characteristics achieved by parallel configurations. It's a common practice in electronics to manage resistance and optimize circuit functionality using this approach of dividing wire into parallel parts to achieve desired outcomes.