Problem 2852 - JEE Physics

Question

Which one of the following is not Ohm's law? \((\mathrm{J}=\) current density, \(\mathrm{E}=\) Electric field, \(\rho=\) resistivity and \(\sigma\) conductivity) (A) \(\mathrm{J}=\sigma \mathrm{E}\) (B) \(\mathrm{J}=\rho \mathrm{E}\) (C) \(\mathrm{I}=(\mathrm{V} / \mathrm{R})\) (D) \(\mathrm{E}=\rho \mathrm{J}\)

Step-by-Step Solution

Answer

The correct answer is (B) J = ρE, as it incorrectly presents the relationship between current density and electric field through resistivity and does not follow Ohm's law.

Step 1: Determine Ohm's Law

Ohm's law states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the points and inversely proportional to the resistance (R) of the conductor. Mathematically, it can be represented as: \[ \text{I} = \frac{\text{V}}{\text{R}} \]

Step 2: Analyze the Options

(A) J = σE Here, J is the current density and E is the electric field. The relationship between current density, electric field, and conductivity (σ) is given by: \[ \text{J} = \sigma \text{E} \] This equation represents Ohm's law in terms of current density, electric field, and conductivity. So this option follows Ohm's law. (B) J = ρE From the equation of resistivity (ρ), \[ \rho = \frac{1}{\sigma} \] Substitute this in the equation given in option A (J = σE): \[ \text{J} = \frac{\text{E}}{\rho} \] This is not Ohm's law, as it incorrectly presents the relationship between current density and electric field through resistivity. (C) I = (\text{V} / \text{R}) This is simply Ohm's law in its most basic form. So, this option follows Ohm's law. (D) E = ρJ The equation for the electric field (E) in terms of resistivity (ρ) and current density (J) can be derived from the previous equations as follows: \[ \text{E} = \rho \text{J} \] This equation represents Ohm's law in terms of electric field, resistivity, and current density. So this option follows Ohm's law.

Step 3: Choose the Correct Answer

From the analysis, it is clear that option (B) J = ρE does not follow Ohm's law. It incorrectly presents the relationship between current density and electric field through resistivity. Therefore, the correct answer is: (B) J = ρE

Key Concepts

Current Density

Current density is a fundamental concept in understanding how electric currents flow within materials. It is denoted by the symbol \( J \) and is defined as the amount of electric charge flowing per unit area of a cross-section. Essentially, it describes how concentrated the flow of charges is at any point in the conductor.

Mathematically, current density is expressed as \( J = \frac{I}{A} \), where \( I \) is the current and \( A \) is the cross-sectional area through which the current flows.

• If \( J \) is high, the current is very concentrated, indicating a higher "density" of moving charges.
• A lower \( J \) value means the charges are more spread out across the conductor’s cross-section.

Current density helps in analyzing more complex circuits and understanding the distribution of electric currents in different materials.

Electric Field

The electric field, represented by the symbol \( E \), describes the force experienced by a charge at any point in space due to other charges. It is a vector quantity, meaning it has both magnitude and direction.

In the context of Ohm's law, the electric field gives us an insight into how voltage (or potential difference) prompts the flow of current through a conductor.

• It can be thought of as the "push" that drives charges to move, creating a current.
• The strength of the electric field determines how strongly the charges are pushed, affecting how much current will flow through a given area.

The relationship between electric field and current density is critical, as seen in Ohm's law: \( \text{E} = \rho \text{J} \) when involving resistivity, which helps in understanding how different materials respond to electric fields.

Resistivity

Resistivity is a material's intrinsic property that quantifies how strongly it opposes the flow of electric current. Represented by the symbol \( \rho \), resistivity is crucial when considering why different materials conduct electricity differently.

A high resistivity means the material does not allow electric current to pass through easily, whereas a low resistivity indicates better conducting properties.

• It is important to note that resistivity is the reciprocal of conductivity \( (\rho = \frac{1}{\sigma}) \).
• Materials like metals typically have low resistivity, making them good conductors.
• Insulators, on the other hand, have high resistivity, hindering the flow of current.

The equation \( \text{E} = \rho \text{J} \) demonstrates how resistivity impacts the relationship between electric field and current density, influencing how charge carriers move within a material.