A 24V battery of internal resistance of 10ohms is connected to a variable resistor. At what value of current drawn from battery the rate of heat produced by the resistor is maximum? Hence find its value.
Delivered power is a maximum when the internal and external resistances are equal, as I recall. Therefore
Rexternal(optimum) = 10 ohms.
Current with that external resistance
= V/Rtotal = 24/20 = 1.2 Amps
Maximum power produced in external resistor =
I^2*Rexternal = (1.2)^2 * 10 = 14.40 W
Check: If I use Rexternal = 11 ohms instead, Rtotal = 21 ohms and I = (24/21) = 1.143 A
I^2 * Rext = 14.37 W
If you use Rexternal = 9 ohms, your will still get a power less than 14.4 W
its not in options
To determine the value of current drawn from the battery at which the rate of heat produced by the resistor is maximum, we need to consider the total resistance in the circuit.
The total resistance in the circuit is the sum of the internal resistance of the battery and the resistance of the variable resistor.
Let's denote the resistance of the variable resistor as R.
The total resistance (R_total) in the circuit can be calculated using the formula:
R_total = internal resistance of battery + resistance of the variable resistor
= 10 ohms + R
Now, let's calculate the current drawn from the battery:
Using Ohm's Law, we have:
Voltage (V) = Current (I) * Resistance (R_total)
In this case, the voltage of the battery is 24V.
Therefore, 24V = I * (10 ohms + R)
To find the value of I that maximizes the rate of heat produced by the resistor, we need to maximize the power (P) dissipated in the resistor.
The power dissipated in the resistor can be calculated using the formula:
P = I^2 * R
To maximize P, we take the derivative of P with respect to I and set it equal to zero:
dP/dI = 2IR - 0 (since we set it equal to zero)
Simplifying the equation:
2IR = 0
Since R cannot be zero (as it represents the resistance of the variable resistor), the only possibility is I = 0.
This means that when no current is drawn from the battery (I = 0), the rate of heat produced by the resistor is maximum.
However, this would not be a practical scenario. In practical situations, the maximum value of current drawn from the battery would be limited by the resistance of the variable resistor and the power capacity of the battery.
Therefore, to calculate the value of current drawn from the battery at which the rate of heat produced by the resistor is maximum, we need more information about the value of the variable resistor (R).
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.