When ‘n’ resistances each of value ‘r’ are connected in parallel, then the resultant resistance is ‘x’. When these ‘n’ resistances are connected in series, total resistance is

Explanation:

Problem Statement:

When 'n' resistances, each of value 'r', are connected in parallel, the resultant resistance is 'x'. When these 'n' resistances are connected in series, the total resistance is to be determined.

Solution:

To solve this problem, we need to calculate the equivalent resistance when the resistances are connected in series. First, let’s analyze the situation for parallel and series connections.

Step 1: Equivalent Resistance in Parallel Connection

When 'n' resistances, each of value 'r', are connected in parallel, the formula for the resultant resistance 'x' is given by:

1/x = 1/r + 1/r + 1/r + ... (n terms)

Since all resistances have the same value 'r', the equation becomes:

1/x = n/r

Rearranging the terms, we get:

x = r/n

This means the equivalent resistance 'x' for the parallel connection is r/n.

Step 2: Equivalent Resistance in Series Connection

When the same 'n' resistances are connected in series, the total resistance is simply the sum of the individual resistances:

Total Resistance = r + r + r + ... (n terms)

This can be written as:

Total Resistance = n × r

Substituting the value of 'r' from the parallel connection equation, where r = x × n (from Step 1), we get:

Total Resistance = n × (x × n) = n² × x

Thus, the total resistance when 'n' resistances are connected in series is n² × x.

Final Answer:

The total resistance in series is n² × x. Therefore, the correct option is Option 1.

Additional Information

To further analyze the problem, let us evaluate the other options:

Option 2: x/n

This option suggests that the total resistance in series is x divided by n. However, this is incorrect because, in a series connection, the resistances add up linearly, and the correct formula is n² × x, not x/n.

Option 3: n²/x

This option suggests that the total resistance in series is n² divided by x. This is incorrect because the formula for series resistance involves multiplying n² with x, not dividing by x.

Option 4: n × x²

This option suggests that the total resistance in series is n multiplied by x². However, this is incorrect as there is no squaring of 'x' in the formula for total resistance in series.

Option 5: Not provided

This option does not contribute to the solution and is irrelevant in this context.

Conclusion:

The correct answer is Option 1: n² × x. This result aligns with the formula for calculating total resistance in a series connection, where the resistances add up linearly, and the relationship between the parallel and series configurations of the resistors is accurately established.