A resistor of 6 ohm is connected in series to a parallel connection of a 30-ohms resistor and a variable resistor R. What must be the value of R such that the power taken by the 6-ohm resistor will be equal to the power taken by the R?
To find the value of resistor R that makes the power taken by the 6-ohm resistor equal to the power taken by R, we first need to understand the formulas for power and how resistors behave in series and parallel connections.
The formula for power (P) is given by P = I^2 * R, where I is the current flowing through the resistor and R is the resistance.
In a series connection of resistors, the current (I) is the same for all resistors, but the total resistance (R_total) is the sum of the individual resistances. Therefore, for the series connection of a 6-ohm resistor and the parallel connection of a 30-ohm resistor and R, we can calculate the total resistance as follows:
R_total = 6 ohm + (30 ohm || R)
In a parallel connection of resistors, the total resistance (R_parallel) can be calculated using the formula:
1/R_parallel = 1/R_1 + 1/R_2
Using this formula, we can rewrite the previous equation as:
1/R_parallel = 1/30 ohm + 1/R
Simplifying further:
1/R_parallel = (R + 30)/(30R)
Taking the reciprocal of both sides, we get:
R_parallel = (30R)/(R + 30)
Now, to find the value of resistor R that makes the power taken by the 6-ohm resistor equal to the power taken by R, we set their respective powers equal to each other:
P_6ohm = P_R
(I^2 * 6) = (I^2 * R_parallel)
Since the currents through both resistors are the same in a series connection, we can cancel out I^2:
6 = R_parallel
Therefore, we set the formula for R_parallel derived earlier equal to 6 ohm:
(30R)/(R + 30) = 6
Now we have to solve this equation for R. To do that, we can cross-multiply and simplify the equation:
30R = 6(R + 30)
30R = 6R + 180
24R = 180
R = 180/24
R = 15/2
So, the value of resistor R that makes the power taken by the 6-ohm resistor equal to the power taken by R is R = 15/2 ohm, which is equivalent to 7.5 ohm.