To analyze the cylindrical element with a current \( I \) flowing through it and a resistivity \( \rho \), we need to consider how the properties of the cylinder affect the electrical characteristics in its two halves. Let's break down each option to determine which one is correct.
Understanding the Basics
First, let's recall some fundamental concepts:
- Resistivity (\( \rho \)): This is a material property that quantifies how strongly a given material opposes the flow of electric current.
- Voltage (\( V \)): This is the potential difference that drives the current through the material.
- Current Density (\( J \)): This is the amount of current flowing per unit area of the cross-section of the cylinder.
- Electric Field (\( E \)): This is the force per unit charge that drives the current through the material.
Evaluating Each Option
Option A: Power loss in first half is four times the power loss in second half.
The power loss in a resistor can be calculated using the formula \( P = I^2 R \). If we assume the resistances of the two halves are equal, the power loss would also be equal. Therefore, this option is incorrect.
Option B: Voltage drop in first half is twice of voltage drop in second half.
For a uniform cylindrical conductor, the voltage drop across each half will be proportional to its resistance. If the resistivity and dimensions are uniform, the voltage drop will be the same across both halves. Thus, this statement is also incorrect.
Option C: Current density in both halves are equal.
Current density \( J \) is defined as \( J = \frac{I}{A} \), where \( A \) is the cross-sectional area. In a uniform cylindrical conductor, the current density remains constant throughout the length of the cylinder. Therefore, this option is correct.
Option D: Electric field in both halves is equal.
The electric field \( E \) in a conductor can be expressed as \( E = \frac{V}{L} \), where \( V \) is the voltage and \( L \) is the length of the conductor. Since the voltage drop is the same across both halves, and the lengths are equal, the electric field will also be equal in both halves. However, this option is not the best choice compared to option C.
Final Thoughts
After analyzing all the options, the correct answer is Option C: Current density in both halves are equal. This conclusion is based on the uniformity of the cylindrical conductor and the principles of current flow in resistive materials.
