Question

The resistance of a conductor is:
A. Inversely proportional to the length.
B. Directly proportional to the square of the radius.
C. Inversely proportional to the square of the radius.
D. Directly proportional to the square root of the length.

Answer 1

Hint: To solve this question, we must know what is resistance and why does resistance occur. In search for a reason as to why resistance occurs we will come across the things on which resistance depends. From the question we can see that it is somehow related to length and radius. Resistance is the property which opposes the flow of current in a material. Resistance is measured in Ohms.

Formula used:
\[I=\dfrac{V}{R}\]

Complete step by step answer:
Resistance is the property of a material which deals with the flow of electricity and therefore, electrons. Resistance opposes the flow of electrons. Resistance is caused when free moving electrons in a conductor collide with the fixed positively charged ions in metal after the voltage is applied. As the length is increased, the number of fixed positive ions increases and thus the collision of free electrons increases. This increases the resistance in a conductor. Thus, resistance is directly proportional to the length. Also, we know that current flows through surface area. Therefore, if the surface area is increased the current has more area to travel which affects the resistance. So, we can say that resistance is inversely proportional to surface area. As surface area directly depends on square of radius, we can now conclude that resistance is inversely proportional to square of radius. Therefore, the correct answer is option C.

Additional Information:
From the above formula known as Ohm’s law resistance depends on voltage applied and current flowing through a resistor. Therefore, we can say that resistance refers to a measure of opposition to the flow of current in a circuit. From Ohm’s law we can derive the formula for resistance which states that the current density vector is \[\overrightarrow{J}\] parallel to the energy field \[\overrightarrow{E}\].
Therefore,
\[\overline{J}=\sigma \overrightarrow{E}=\dfrac{1}{\rho }\overrightarrow{E}\] , where \[\rho \]is the resistivity…… (1)
\[\overrightarrow{E=}\dfrac{\Delta V}{L}\]……. (2)
Also, we can express current flowing as,
\[\overrightarrow{J}=\dfrac{I}{A}\]……. (3)
From, (1), (2) and (3),
\[\dfrac{I}{A}=\dfrac{1}{\rho }\dfrac{\Delta V}{L}\]
On rearranging the terms,
\[\Delta V=I\dfrac{\rho L}{A}\]
Therefore, we can say that resistance is directly proportional to Length and inversely proportional to the area.

Note:
The above derivations are quite complex in nature, students might make mistakes while solving. It is therefore better to understand the concept in theory first. Resistance causes some of the energy to be lost as heat energy. Therefore, resistance can be said to be good and bad. If we are trying to conduct electricity, then resistance is said to be bad in a conductor. However, if our light bulbs work on the principle of electric energy converted into heat energy. So, in this case more resistance is more valuable it will be.