To address the question of which factor the motional electromotive force (emf) induced in a coil is independent of, it's important to understand the principle behind motional emf first.
The motional emf in a coil is described by Faraday's Law of Electromagnetic Induction. According to this law, the induced emf in a coil is given by:
[tex]\[ \text{emf} = -N \frac{d\Phi}{dt} \][/tex]
Here:
- [tex]\( \text{emf} \)[/tex] is the induced electromotive force,
- [tex]\( N \)[/tex] is the number of turns in the coil,
- [tex]\( \frac{d\Phi}{dt} \)[/tex] is the rate of change of magnetic flux [tex]\( \Phi \)[/tex] through the coil.
Breaking down the components:
- Number of turns, [tex]\( N \)[/tex]: This directly affects the magnitude of the induced emf. More turns will result in a greater emf by multiplying the rate of change of magnetic flux.
- Change in time, [tex]\( t \)[/tex]: The rate of change of magnetic flux [tex]\( \frac{d\Phi}{dt} \)[/tex] inherently contains the element of time. Thus, time plays a crucial role.
- Change in magnetic flux, [tex]\( \Phi \)[/tex]: The emf is directly dependent on how the magnetic flux through the coil changes over time. Therefore, any change in magnetic flux influences the emf.
Now, addressing the fourth factor:
- Resistance, [tex]\( R \)[/tex]: The resistance is not present in Faraday's Law equation for induced emf. Resistance affects the current flowing through the circuit once the emf has been induced, but it has no effect on the emf itself.
Given our analysis, we can conclude that the motional emf is independent of the resistance in the circuit.
Thus, the correct answer is:
D. Resistance
