A certain number of identical resistors are connected in series and their total resistance is measured. When the resistors are disconnected and re-connected in parallel, their total resistance drops to one-percent of the value measured when they were connected in series. How many resistors are there?
For this question, I know the answer. What I need help with is finding assumptions and/or simplifications:
Determine which physics concepts and/or laws are involved and what assumptions and/or simplifications can be made about the physical situation in order to apply them. What simplifications are reasonable? The assumptions which simplify the problem must be explicitly stated and consistent with the applicable concepts. For example, if momentum conservation is being applied to a system of two cars in a collision, then friction from the road will be ignored since it is an external force and the system has been simplified to have no external forces in order to apply the conservation law.
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To solve this problem, we need to apply the concepts of resistance in series and parallel circuits.
Assumptions:
1. The resistors are identical, which means they have the same resistance value.
2. The resistors are perfectly connected in series and parallel.
3. The resistors have negligible internal resistance.
Simplifications:
1. We will assume that the wires connecting the resistors have negligible resistance.
2. We will ignore any other components in the circuit that may affect the overall resistance.
3. We will consider only the ideal behavior of the resistors according to Ohm's Law, where resistance remains constant with variation in current and voltage.
With these assumptions and simplifications in mind, we can now solve the problem.
Let's assume the resistance of each resistor is R.
When resistors are connected in series, their total resistance adds up. So, if there are n resistors in series, the total resistance (Rs) can be calculated as:
Rs = n * R (Equation 1)
When resistors are connected in parallel, the total resistance reduces according to the formula:
1/Rp = (1/R) + (1/R) + ... + (1/R) (Equation 2)
= n/R
Rearranging Equation 2, we get:
Rp = R/n (Equation 3)
Given that the total resistance in parallel is one-percent (1/100) of the total resistance in series, we can equate Equation 3 to Equation 1:
Rp = Rs/100
Substituting Rs = n * R and solving for n:
R/n = (n * R)/100
Cross-multiplying:
100 * R = n^2 * R
Dividing both sides by R:
100 = n^2
Taking the square root of both sides:
n = √100 = 10
Therefore, there are 10 resistors in the circuit.