A non-linear resistor has the characteristic i = k * v^4. If the current i increases to 100 times its original value, by what factor does the voltage v change?

Introduction / Context:This question checks understanding of power law device characteristics and how input and output variables scale. Many non-linear components exhibit current that is a power function of voltage. Knowing how one variable scales when the other is multiplied is a key skill in electronics and circuit analysis.

Given Data / Assumptions:

• The device obeys i = k * v^4, where k is a constant of proportionality.
• The current increases by a factor of 100 (i_new = 100 * i_old).
• We assume the same operating device and temperature, so k remains constant.

Concept / Approach:

If i ∝ v^n, then v ∝ i^(1/n). Here n = 4, so voltage scales with the fourth root of current. We translate the 100x increase in current into a voltage scaling using the inverse power 1/4. This method avoids needing the exact value of k or absolute voltages and currents.

Step-by-Step Solution:

Verification / Alternative check:

Plug back: If v scales by 3.162, then v^4 scales by (3.162)^4 ≈ 100, matching the required current change. This confirms the reasoning without needing k.

Why Other Options Are Wrong:

• about 10 times and about 100 times: imply v ∝ i or v ∝ i^something much larger; both contradict the fourth-power law.
• about twice or about √2 times: far too small; would yield current increases of 2^4 = 16 or (√2)^4 = 4, not 100.

Common Pitfalls:

• Taking a square root instead of the fourth root because of confusion between exponent and root.
• Assuming linear proportionality, which would be incorrect for power-law devices.

Final Answer:

about 3 times