To find the value of the current at which the rate of heat produced by the resistor is maximum, we need to maximize the power dissipated by the resistor.
Power (P) dissipated by a resistor can be calculated using the formula:
P = (I^2) * R
Where:
P is the power dissipated by the resistor,
I is the current flowing through the resistor, and
R is the resistance of the resistor.
Given that the battery voltage is 24V, and the internal resistance is 10 ohms, the total resistance in the circuit (including the variable resistor) can be written as:
Total resistance (R_total) = R_variable + R_internal
To find the rate of heat produced by the resistor, we need to calculate the power dissipated by the resistor, which is equal to the power supplied by the battery. Therefore, we can write:
Power supplied by the battery = P = V * I
Where the voltage (V) is the battery voltage.
Since the internal resistance is always present in the circuit, the current flowing through the circuit (I_total) is given by:
I_total = V / R_total
Substituting the value of R_total, we have:
I_total = V / (R_variable + R_internal)
Now, to maximize the power dissipated by the resistor, we differentiate the power with respect to I and set it to zero.
dP/dI = 2 * I * R_variable - V = 0
Simplifying the equation, we have:
2 * I * R_variable = V
Solving for I, we get:
I = V / (2 * R_variable)
Now, let's substitute the given values into the equation:
V = 24V
R_internal = 10 ohms
I = 24V / (2 * R_variable)
Finally, we can calculate the value of the current (I) at which the rate of heat produced by the resistor is maximum by substituting the given values into the equation.