Resistance And Resistivity

For the power delivered to the load resistor in each case, we use the power formula: If the power is the same for both resistors, we equate the powers for both configurations: Solving this equation gives the internal resistance .

Step 1: Understanding the Concept:
This is a multi-step problem that connects the electrical properties of a wire (resistance, resistivity) with its geometrical properties (length, area) and how it's shaped into a coil (number of turns, radius). We first need to find the total length of the wire using the resistance formula and then use that length to find the number of turns in the coil.

Step 2: Key Formula or Approach:
1. Resistance of a wire: , where is resistance, is resistivity, is length, and is the cross-sectional area. 2. Cross-sectional area of the wire: , where is the radius of the wire. 3. Total length of the wire in the coil: , where is the number of turns and the circumference is .

Step 3: Detailed Explanation:
Given data:
Total resistance, .
Coil radius, .
Wire diameter, , so wire radius, .
Resistivity, .
Calculation:
Part 1: Find the total length (L) of the wire.
First, calculate the cross-sectional area ( ) of the wire: Now, use the resistance formula to find the length : Part 2: Find the number of turns (n).
The length of the wire required for one turn is the circumference of the coil: The total number of turns is the total length of the wire divided by the length of one turn:

Step 4: Final Answer:
The number of turns in the coil will be 70.

Step 1: Understanding the Concept:
This question asks about the factors that determine the electrical resistivity ( ) of a conducting material. It is important to distinguish between resistance ( ) and resistivity ( ).

Step 2: Detailed Explanation:
Resistivity ( ):
Resistivity is an intrinsic property of a material. It quantifies how strongly a given material opposes the flow of electric current.
- Material: Different materials have different atomic structures and numbers of free charge carriers per unit volume. For example, copper has a lower resistivity than iron because it has more free electrons available for conduction. Therefore, resistivity depends on the material.
- Temperature: When the temperature of a conductor increases, the atoms/ions in the material vibrate with greater amplitude. This increases the frequency of collisions between the free electrons and the ions, which impedes the flow of electrons and thus increases the resistivity. Therefore, resistivity depends on temperature.
Resistance ( ):
Resistance is an extrinsic property of an object. It depends not only on the material's resistivity but also on the object's physical dimensions (length and cross-sectional area ). The relationship is given by: From this formula, we can see that resistance depends on dimensions, but resistivity ( ) does not.
Evaluating the options:
(A) Incorrect. Resistivity does not depend on the dimensions.
(B) Correct. Resistivity depends on the nature of the material and its temperature.
(C) Incorrect. Resistivity does not depend on dimensions.
(D) Incorrect. While it depends on temperature, it also fundamentally depends on the material itself.

Step 3: Final Answer:
The resistivity of a conductor is an intrinsic property that depends on the nature of the material and its temperature.