The back emf in a DC motor is maximum when,

Options

A. Motor has picked up maximum speed
B. Motor has just started moving
C. Speed of motor is still increasing
D. Motor has been switched off

Answer

To determine when the back emf in a DC motor is maximum, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Back EMF**:
Back emf (electromotive force) in a DC motor is the induced emf that opposes the applied voltage. It is generated due to the rotation of the motor's coil in a magnetic field.

2. **Relation to Speed**:
The back emf is directly related to the speed of the motor. As the motor speeds up, the rate of change of magnetic flux through the coil increases, which in turn increases the back emf.

3. **Magnetic Flux**:
The magnetic flux (Φ) through the coil is given by the formula:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
where:
- \(B\) is the magnetic field strength,
- \(A\) is the area of the coil,
- \(\theta\) is the angle between the magnetic field and the normal to the area of the coil.

4. **Rate of Change of Magnetic Flux**:
The induced emf (E) can be expressed as:
\[ E = -\frac{d\Phi}{dt} \]
To maximize the induced emf, we need to maximize the rate of change of magnetic flux (\(\frac{d\Phi}{dt}\)).

5. **Differentiating Magnetic Flux**:
When the coil rotates, \(\theta\) changes, and thus:
\[ \frac{d\Phi}{dt} = B \cdot A \cdot \frac{d(\cos(\theta))}{dt} \]
Using the chain rule, we have:
\[ \frac{d(\cos(\theta))}{dt} = -\sin(\theta) \cdot \frac{d\theta}{dt} \]
Therefore:
\[ \frac{d\Phi}{dt} = -B \cdot A \cdot \sin(\theta) \cdot \frac{d\theta}{dt} \]

6. **Maximum Rate of Change**:
The term \(\frac{d\theta}{dt}\) represents the angular speed (\(\omega\)) of the motor. The back emf is maximum when \(\omega\) is at its maximum value, which occurs when the motor has reached its maximum speed.

7. **Conclusion**:
Thus, the back emf in a DC motor is maximum when the motor has picked up maximum speed.

### Final Answer:
The back emf in a DC motor is maximum when the motor has picked up maximum speed.

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