How Do Resistors in Series and Parallel Affect Circuit Behavior?

[stonnn]

i really have no clue how to do these. they seem more conceptual. any hints would be welcome! thanks!

1. Prove that when n identical resistors of resistance R are connected in parallel the equivalent resistance is R/n.

2. A piece of copper wire of resistance R is cut into 3 equal parts. When these 3 parts are connected in parallel, what is the resistance of the combination?

3. Say you have a series and parallel circuit. Resistors 2 and 3 are parallel to each other. Resistor 1 is in series with them. When you increase the resistance of 3, what happens to current and voltage values?

Thanks!

 

[lanedance]

hi stonn
do you know the equations for combining resistance in parallel & series?

 

[nlightner]

I understand that series and parallel circuits can be confusing at first, but with some basic principles and equations, we can easily solve these types of questions. Let's take a look at each of the questions and see how we can approach them.

1. To prove that when n identical resistors of resistance R are connected in parallel, the equivalent resistance is R/n, we can use the formula for parallel resistors, which states that the equivalent resistance (Req) is equal to the reciprocal of the sum of the reciprocals of each individual resistance. Mathematically, this can be represented as:

1/Req = 1/R + 1/R + ... + 1/R (n times)

Simplifying this equation, we get:

1/Req = n/R

Multiplying both sides by R, we get:

Req = R/n

Therefore, we have proven that the equivalent resistance in a parallel circuit with n identical resistors is R/n.

2. In this question, we are given a copper wire of resistance R that is cut into 3 equal parts and then connected in parallel. To find the resistance of the combination, we can use the same formula as in question 1. Since the three parts are connected in parallel, their equivalent resistance would be:

Req = R/3

Therefore, the resistance of the combination is R/3.

3. In a series and parallel circuit, the current remains the same throughout the circuit while the voltage is divided among the components. In this scenario, resistor 1 is in series with resistors 2 and 3, which are in parallel with each other. When we increase the resistance of resistor 3, it will cause an increase in the total resistance of the parallel branch. This, in turn, will lead to a decrease in the overall current in the circuit, as per Ohm's Law (V=IR). As for the voltage values, they will remain the same across each resistor in the parallel branch, but the voltage across resistor 1 will increase as the total current decreases.

I hope this helps give you some direction in solving these types of questions. Remember to use the appropriate formulas and principles, and don't be afraid to break down the circuit into smaller parts to better understand its behavior. Good luck!