The Characteristics of a Non-Linear Resistance:

A non-linear resistance is a type of resistance that does not follow Ohm's Law, which states that the current passing through a conductor is directly proportional to the voltage applied across it. In the case of a non-linear resistance, the relationship between current (I) and voltage (V) is not a straight line, but rather a curve. One example of a non-linear resistance is a diode.

The Given Relationship:

In the given question, the relationship between current and voltage is given as: i = kv^4, where i is the current and v is the voltage. This equation shows that the current is proportional to the fourth power of the voltage.

Understanding the Relationship:

To understand how the current and voltage are related in this non-linear resistance, let's consider an example. If the voltage across the non-linear resistance is doubled, i.e., v' = 2v, then the current can be calculated as follows:

i' = k(2v)^4
= 16k(v^4)

This shows that when the voltage is doubled, the current becomes 16 times the original value. Similarly, if the voltage is tripled (v' = 3v), then the current becomes:

i'' = k(3v)^4
= 81k(v^4)

This indicates that when the voltage is tripled, the current becomes 81 times the original value.

Relationship between Current and Voltage:

From the above calculations, we can observe that when the voltage is increased by a factor of n, the current becomes n^4 times the original value. In other words, the current is proportional to the fourth power of the voltage.

Answer Explanation:

In the given question, it is mentioned that the current becomes 100 times. Using the relationship between current and voltage in a non-linear resistance, we can calculate the factor by which the voltage changes as:

100 = k(v^4)
v^4 = 100/k

To find the factor by which the voltage changes, we can take the fourth root of both sides:

v = (100/k)^(1/4)

Since k is a constant, (100/k) is also a constant. Therefore, the fourth root of a constant is also a constant. This means that when the current becomes 100 times, the voltage becomes a constant factor, which is approximately 3 times.

Therefore, the correct answer is option 'C': about 3 times.