What is the non-linear $V-I$ relation in a P-N junction in forward bias?
A) $I={{I}_{s}}{{e}^{V/{{V}_{T}}}}$
B) $I={{I}_{s}}({{e}^{V/{{V}_{T}}}}-1)$
C) $I={{I}_{s}}(1-{{e}^{V/{{V}_{T}}}})$
D) $I={{I}_{s}}{{(\dfrac{V}{{{V}_{0}}})}^{\dfrac{3}{2}}}$

Hint: In case of forward bias of a P-N junction, we know, the current depends on the applied voltage exponentially. In this relation, there is an ideality factor term $\eta$ . For germanium, the value of this term is $1$ , and so by putting this value in the above relation, we can get our required non-liner $V-I$ relation.

Complete answer:
The voltage-current characteristics of an electrical device is known as the $V-I$ characteristics.
As the resistance of the P-N junction diode gets changed according to the direction of current flow, just above the zero bias, hence it is referred to as a non-linear instrument. “Linear” means the voltage and current are always proportional. Unlike a resistor, the diode does not act linearly with respect to the applied voltage because of the diode having an exponential $V-I$ relationship. This exponential relation between voltage and current can be expressed as,
$\Rightarrow I={{I}_{s}}({{e}^{\dfrac{V}{\eta {{V}_{T}}}}}-1)$ .
Where, ${{I}_{s}}$ is the reverse saturation current,
$\eta$ is the (exponential) ideality factor,
$V$ is the voltage applied,
${{V}_{T}}$ is the voltage equivalent of the temperature.
We know that, value of $\eta$ is $1$ for germanium, and $2$ for silicon.
Therefore, for germanium diode, as $\eta =1$ ,
The non-linear $V-I$ relation in forward bias is, $I={{I}_{s}}({{e}^{V/{{V}_{T}}}}-1)$.

Therefore, the correct answer is (B), $I={{I}_{s}}({{e}^{V/{{V}_{T}}}}-1)$.

Additional information:
Using the opposite kind of doping, if one part of a semiconducting crystal is made n-type and the other part p-type, then that crystal is known as a P-N junction. But, by joining two different p-type and n-type crystals, a P-N junction is not formed because in that case, the crystals would not be joined uniformly and so the junction would not act properly.

Note: When the p-end is connected with the positive terminal of the external source of electricity, and n-end with the negative terminal, we can say, forward bias is applied to a P-N junction. And when n-end is connected with the positive terminal of source and p-end is connected with the negative terminal, the P-N junction is said to be in reversed biased condition.