Question

A steady current flow in a metallic conductor of non-uniform crosssection. Which of these quantities is constant along the conductor: current, current density, electric field, drift speed?

Answer 1

Here is a detailed explanation for each quantity:

1. Current (I)

• Why it's constant: The term "steady current" implies that the rate of flow of charge ($I = dQ/dt$) is constant. Due to the principle of conservation of charge, charge cannot be created, destroyed, or accumulated at any point within the conductor in a steady state. Therefore, the same amount of charge that enters any cross-section of the conductor per second must also exit any other cross-section in the same amount of time.
• Conclusion: Current (I) is constant throughout the entire length of the conductor.

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2. Current Density (J)

• Why it's not constant: Current density is defined as the current per unit cross-sectional area ($J = I/A$).
  • We have already established that the current (I) is constant.
  • The problem states that the conductor has a non-uniform cross-section, which means the area (A) changes along its length.
• Conclusion: Since $J = (\text{constant}) / (\text{variable})$, the current density (J) is not constant. It will be smaller in the wider sections of the conductor and larger in the narrower sections.

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3. Electric Field (E)

• Why it's not constant: The electric field is related to the current density by the microscopic form of Ohm's Law: $E = \rho J$, where $\rho$ (rho) is the resistivity of the material.
  • Assuming the conductor is made of the same metallic material throughout, its resistivity ($\rho$) is constant.
  • We have just shown that the current density (J) is not constant.
• Conclusion: Since $E = (\text{constant}) \times (\text{variable})$, the electric field (E) is not constant. A stronger electric field is required in the narrower sections (where J is high) to drive the current.

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4. Drift Speed ($v_d$)

• Why it's not constant: The relationship between current and the drift speed of charge carriers (electrons in a metal) is given by the equation: $I = nAev_d$.
  • $I$ = current (constant)
  • $n$ = number of charge carriers per unit volume (constant for a given material)
  • $A$ = cross-sectional area (variable)
  • $e$ = elementary charge (a fundamental constant)
  • $v_d$ = drift speed
• Rearranging for drift speed: $v_d = \frac{I}{nAe}$.
• Conclusion: Since $v_d = (\text{constant}) / (\text{variable})$, the drift speed ($v_d$) is not constant. The electrons will move more slowly in the wider sections (large A) and speed up in the narrower sections (small A).

Summary

| Quantity | Constant or Not? | Reason |
| :--- | :--- | :--- |
| Current (I) | Constant | Conservation of charge in a steady flow. |
| Current Density (J) | Not Constant | $J = I/A$. Since A is not constant, J is not constant. |
| Electric Field (E) | Not Constant | $E = \rho J$. Since J is not constant, E is not constant. |
| Drift Speed ($v_d$) | Not Constant | $v_d = I/(nAe)$. Since A is not constant, $v_d$ is not constant. |