Joule's law of heating states that the heat produced in a conductor is directly proportional to the square of the current passing through it, the resistance of the conductor, and the time for which the current flows. Mathematically, it can be expressed as: H=I ^2 Rt .
Here are two ways to join three resistors, each of resistance 9 Ω, so that the equivalent resistance of the combination is 13.5 Ω:
Method 1:
Connect the resistors in series. The equivalent resistance of two resistors in series is the sum of their individual resistances.
Therefore, the equivalent resistance of three 9 Ω resistors in series is 9 Ω + 9 Ω + 9 Ω = 27 Ω.
However, we can reduce this resistance to 13.5 Ω by connecting a 4.5 Ω resistor in parallel with one of the 9 Ω resistors.
The equivalent resistance of two resistors in parallel is given by the following formula:
Req = (R1 * R2) / (R1 + R2)
where Req is the equivalent resistance, R1 and R2 are the individual resistances. Plugging in the values of R1 = 9 Ω and R2 = 4.5 Ω, we get:
Req = (9 Ω * 4.5 Ω) / (9 Ω + 4.5 Ω) = 13.5 Ω
Therefore, the equivalent resistance of the combination is 13.5 Ω.
Method 2:
Connect the resistors in a parallel-series combination. A parallel-series combination is a combination of resistors in which some resistors are connected in parallel and the remaining resistors are connected in series with the parallel combination.
The equivalent resistance of a parallel-series combination can be calculated using the following formula:
Req = (R1 * R2 * R3) / (R1 * R2 + R2 * R3 + R3 * R1)
where Req is the equivalent resistance, R1, R2, and R3 are the individual resistances. Plugging in the values of R1 = 9 Ω, R2 = 9 Ω, and R3 = 9 Ω, we get:
Req = (9 Ω * 9 Ω * 9 Ω) / (9 Ω * 9 Ω + 9 Ω * 9 Ω + 9 Ω * 9 Ω) = 13.5 Ω
Therefore, the equivalent resistance of the combination is 13.5 Ω.
(a) Joule's law of heating states that the heat generated by an electric current in a conductor is directly proportional to the square of the current, the resistance of the conductor, and the time for which the current flows.
The formula for Joule's law is:
H = I^2 * R * t
where H is the heat generated in joules, I is the current in amperes, R is the resistance in ohms, and t is the time in seconds.