A piece of wire of resistance \(R\) is cut into five equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is \(R^{\prime}\), then the ratio \(R / R^{\prime}\) is - (a) \(1 / 25\) (b) \(1 / 5\) (c) 5 (d) 25

Ohm's Law is fundamental in understanding the relationship between voltage (V), current (I), and resistance (R) in electrical circuits. The law is typically represented by the equation: \[ V = IR \]
This tells us that the voltage across a resistor is equal to the product of the current flowing through it and the resistance of the resistor.
Knowing Ohm's Law is crucial because it underpins much of the theory in electrical circuits.

• Voltage (V): The potential difference between two points in a circuit. Measured in volts (V).
• Current (I): The rate of flow of electric charge. Measured in amperes (A).
• Resistance (R): The opposition to the flow of current. Measured in ohms (Ω).

This principle helps students understand the basics before jumping into more complex concepts like series and parallel circuits.

Series and parallel circuits describe two different ways in which resistors (and other components) can be connected in a circuit. Each type of circuit has unique characteristics that affect the overall resistance and behavior of the circuit.
In a **series circuit**, resistors are connected end-to-end, so the same current flows through each resistor. The total resistance is simply the sum of the individual resistances: \[ R_{total} = R_1 + R_2 + R_3 + \text{...} \]
In a **parallel circuit**, resistors are connected across the same two points, providing multiple paths for the current. The total resistance in a parallel circuit is found using the formula: \[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \text{...} \]
These concepts are vital for understanding how resistors behave when combined in different configurations. Knowing this helps in solving problems like the given exercise, where resistors were connected in parallel after being cut from a single wire.

Calculating electrical resistance is key to understanding how circuits behave under different configurations. Electrical resistance depends on the material, length, and cross-sectional area of the conductor. In the given exercise, we determined the equivalent resistance of resistors connected in parallel.
Here’s a step-by-step simplification for parallel resistors:

• First, break down each segment of resistance. If you have a wire with resistance \( R \) cut into five equal parts, each part will have a resistance of \( R/5 \).
• Next, apply the parallel resistance formula: \[ \frac{1}{R'} = \frac{1}{R/5} + \frac{1}{R/5} + \frac{1}{R/5} + \frac{1}{R/5} + \frac{1}{R/5} \]
• Simplify the equation: \[ \frac{1}{R'} = 5 \times \frac{1}{R/5} = 5 \times \frac{5}{R} = \frac{25}{R} \]
• Hence, the equivalent resistance \( R' \) is: \[ R' = \frac{R}{25} \]
• Finally, finding the ratio results in: \[ \frac{R}{R'} = 25 \]

Understanding these calculations clarifies how resistances combine in different configurations, which is essential for designing and analyzing electrical circuits.