Formula: Induced Voltage and the Definition of Inductance
$$U ~=~ - L \, \frac{\text{d} \class{red}{I}}{\text{d} t}$$
$$U ~=~ - L \, \frac{\text{d} \class{red}{I}}{\text{d} t}$$
$$\frac{\text{d} \class{red}{I}}{\text{d} t} ~=~ -\frac{U}{L}$$
$$L ~=~ -\frac{U}{ \dot{\class{red}{I}} }$$
Induced voltage
$$ U $$ Unit $$ \mathrm{V} = \frac{ \mathrm J }{ \mathrm C } = \frac{ \mathrm{kg} \, \mathrm{m}^2 }{ \mathrm{A} \, \mathrm{s}^3 } $$
Induced voltage is the electrical voltage generated between the ends of the coil due to the change of the coil current over time.
The minus sign in the formula takes into account the Lenz law and states that the induced voltage opposes the current change. Nature tries to counteract the change in current.
Current change
$$ \dot{\class{red}{I}} $$
Current change is the derivative of the electric current \(I(t)\) through the coil with respect to time \(t\). A change in current in the coil can be generated, for example, by applying an AC voltage. The faster the current changes, the greater the induced voltage.
The unit is amperes per second. And the time derivative of the current is sometimes compactly noted in physics with a dot over the \( \class{red}{I} \): \(\dot{\class{red}{I}}\).
Inductance
$$ L $$ Unit $$ \mathrm{H} = \frac{ \mathrm{Vs} }{ \mathrm{A} } $$
Inductance is a property of the coil and tells how much "resistance" the coil exerts to the current change. In this equation, it is the proportionality constant between the induction voltage \(U\) and the current change \(\frac{\text{d} I}{\text{d} t}\).