The electrical motor runs on a D.C source of emf E and internal resistance r. Show that power output of the motor is max where current drawn by the motor is E/2r.
To show that the power output of the motor is maximized when the current drawn by the motor is E/2r, we can use the formula for power (P) in an electrical circuit.
The power (P) in an electrical circuit can be calculated using the formula:
P = I * V
Where:
P is the power output
I is the current drawn by the motor
V is the voltage across the motor
In this case, the voltage across the motor can be represented as:
V = E - (I * r)
Since the voltage across the motor depends on the current drawn by the motor and the internal resistance, we can substitute this expression into the formula for power:
P = I * (E - (I * r))
Now, to find the maximum power output, we need to find the value of I that maximizes P. To do that, we can differentiate the expression for P with respect to I and set it equal to zero:
dP/dI = E - 2Ir = 0
Solving for I:
E = 2Ir
I = E/2r
Substituting this value of I back into the equation for power:
P = (E/2r) * (E - (E/2r) * r)
Simplifying this expression, we get:
P = (E^2)/(4r)
Therefore, the power output of the motor is maximized when the current drawn by the motor is E/2r.
To summarize:
1. Start with the formula for power: P = I * V
2. Substitute the expression for voltage across the motor: V = E - (I * r)
3. Substitute this expression back into the power formula: P = I * (E - (I * r))
4. Differentiate P with respect to I and set it equal to zero: dP/dI = E - 2Ir = 0
5. Solve for I: I = E/2r
6. Substituting this value of I back into the power formula: P = (E/2r) * (E - (E/2r) * r)
7. Simplify the expression to get the final result: P = (E^2)/(4r)
Hence, the maximum power output of the motor occurs when the current drawn by the motor is E/2r.