A wire of length L and diameter D has a resistance R.
If BOTH the length and the diameter of the same wire are decreased to half of their original values, the resistivity of the new wire:

A. doubles.
B. drops to half its original value.
C. quadruples
D. drops to one quarter its original value.
E. remains the same.

There seems to be a typo in the question. The question asks to find the resistivity, instead of resistance. Resistivity is a property of the wire and does not change when its physical dimensions change.

(A) doubles

Explanation:

The resistivity of a wire is defined by

R is the resistance, A is the cross-sectional area and l is the length of the wire.

It follows that

Assuming a circular cross section, the area is given by

where D is the diameter.

Then

k is an arbitrary constant

When l and D are halved, the new resistance, R1 is

Hence, the resistance is doubled.

adding a(n) to an electric circuit would increase the magnetic field around a wire;

b. solenoid

a solenoid is a type of electromagnet whose purpose is to generate a controlled magnetic field.

if the purpose of the solenoid is instead to impede changes in the electric current, a solenoid can be more specifically classified as an inductor rather than an electromagnet.

the higher the current the higher the mag field around it the higher the influence on the compass