Current Density

Given current density:
To find the current from to , we integrate over the circular cross-section in cylindrical coordinates: So, the current is proportional to . Final answer:

We have three charges at the vertices of a triangle ABC. The charge at vertex A is , and the charges at vertices B and C are . The net electric field at the midpoint M of the base BC is the vector sum of the electric fields due to each charge. - The electric field due to the charge at B and C will cancel each other out because of symmetry.
- The electric field due to the charge at A will point along the line AM (because the distance from A to M is along the line connecting A and M).
The magnitude of the electric field at M due to the charge at A is given by Coulomb's Law: Thus, the net electric field is: Final Answer: (C) pointing along AM

Displacement current:
The displacement current was introduced by James Clerk Maxwell to explain the continuity of current in circuits involving capacitors. It is given by the rate of change of electric flux through a given surface. This term is necessary to maintain the symmetry of Ampere's law for circuits with capacitors, where no physical current flows through the capacitor plates but the current still appears in the circuit. Mathematically, the displacement current is given by: where is the permittivity of free space, and is the electric flux, which is the product of the electric field and the area through which it passes. For a capacitor being charged by a battery, the current flowing into the capacitor is the same as the displacement current between the plates of the capacitor. This current does not flow through the dielectric but is a result of the changing electric field. Ampere-Maxwell law:
The Ampere-Maxwell law states: where is the conduction current, and the second term represents the displacement current . This law ensures the continuity of current because the displacement current term accounts for the changing electric field in the capacitor, which gives rise to a "current" flowing through the dielectric, ensuring that the current remains continuous. Since the total current remains constant in the circuit, we have: Thus, the continuity and constancy of the current are justified by the Ampere-Maxwell law, even in the presence of a capacitor.

Concept: Current in a conductor is due to drift of free electrons under an electric field. Key relations:

Drift velocity:
Current density:

Step 1: Expression for current. If = number of free electrons per unit volume, Charge passing per second through area : Since electron charge magnitude = :
Step 2: Current density.
Step 3: Substitute drift velocity.
Step 4: Compare with Ohm’s law (microscopic form). Comparing: Final Result: Conclusion: Conductivity depends on:

Number of charge carriers
Relaxation time
Electron charge and mass