A 72-ohm coil of wire is connected in series with an adjustable resistor (rheostat) whose resistance can be varied from 0 to 88 ohms. If the line potential is 115 volts, calculate (a) the range through which the coil current can be varied, (b) the rheostat resistance when the power taken by the coil is 90 watts
a is right on both accounts but I don't get your answer for b.
P = E*I
90/115 = I
R = E/I = 115/I from above. I get about 147 ohms and subtract 72 from that.
i've gotten (a) 0.72 to 1.6 amp; (b)17.7 ohms
To calculate the range through which the coil current can be varied and the rheostat resistance when the power taken by the coil is 90 watts, we need to use the formulas for power and resistance in a series circuit.
(a) The range through which the coil current can be varied is determined by the range of the rheostat resistance. Since the coil resistance is fixed at 72 ohms, the total resistance in the circuit will be the sum of the coil resistance and the rheostat resistance.
Total resistance (R) = coil resistance + rheostat resistance
Given:
Coil resistance (Rc) = 72 ohms
Rheostat resistance (Rr) = 0 to 88 ohms (adjustable range)
The total resistance can be varied from 72 ohms (when rheostat resistance is 0 ohms) to 72 + 88 = 160 ohms (when rheostat resistance is 88 ohms).
Therefore, the range through which the coil current can be varied is from 115V / 72 ohms to 115V / 160 ohms.
(b) The power taken by the coil can be calculated using the formula:
Power (P) = (Current (I))^2 * Resistance (Rc)
Given:
Power (P) = 90 watts
Coil resistance (Rc) = 72 ohms
Rearranging the formula:
Current (I) = sqrt(P / Rc)
Substituting the given values:
Current (I) = sqrt(90 watts / 72 ohms)
Calculating the square root:
Current (I) = sqrt(1.25 A^2) approximately 1.12 A
To find the rheostat resistance (Rr) when the power taken by the coil is 90 watts, we can use the formula:
Total resistance (R) = coil resistance (Rc) + rheostat resistance (Rr)
The total resistance is calculated using the formula:
R = (Voltage (V))^2 / Power (P)
Substituting the given values:
R = (115V)^2 / 90 watts
Calculating the resistance:
R = 148.47 ohms
Since the total resistance (R) is the sum of the coil resistance (Rc) and the rheostat resistance (Rr), we can rearrange the formula as follows:
Rheostat resistance (Rr) = Total resistance (R) - Coil resistance (Rc)
Substituting the values:
Rheostat resistance (Rr) = 148.47 ohms - 72 ohms
Calculating the rheostat resistance:
Rheostat resistance (Rr) = 76.47 ohms
Therefore, the rheostat resistance when the power taken by the coil is 90 watts is approximately 76.47 ohms.
To calculate the range through which the coil current can be varied and the rheostat resistance when the power taken by the coil is 90 watts, we need to apply Ohm's Law and power formulas.
(a) The range through which the coil current can be varied:
1. Start by finding the maximum and minimum values of total resistance in the circuit.
Maximum resistance:
Total resistance = coil resistance + rheostat resistance
The coil resistance is given as 72 ohms.
The maximum rheostat resistance is given as 88 ohms.
Maximum total resistance = 72 ohms + 88 ohms = 160 ohms
Minimum resistance:
The minimum rheostat resistance is given as 0 ohms, as the rheostat's resistance can be varied from 0 to 88 ohms.
Minimum total resistance = 72 ohms + 0 ohms = 72 ohms
2. Next, calculate the maximum and minimum values of current using Ohm's Law (I = V / R).
Maximum current:
Using the maximum total resistance, the equation becomes:
I_max = 115 volts / 160 ohms
Minimum current:
Using the minimum total resistance, the equation becomes:
I_min = 115 volts / 72 ohms
(a) The range in which coil current can be varied is from I_min to I_max.
(b) The rheostat resistance when the power taken by the coil is 90 watts:
1. We can determine the power using the formula P = I^2 * R, where P is power, I is current, and R is resistance.
Given that the power is 90 watts and the coil resistance is 72 ohms, we can rearrange the formula:
90 watts = I^2 * 72 ohms
I^2 = 90 watts / 72 ohms
I^2 = 1.25
2. Calculate the current by taking the square root of 1.25:
I = √1.25
Then, determine the rheostat resistance using Ohm's Law:
Rheostat resistance = (Line potential - Coil resistance * I) / I
Substitute the values into the formula to find the rheostat resistance.