A steady current flows in a metallic conductor of non-uniform cross-section. The quantity/quantities

Q: Why doesn't the current increase when the wire gets narrower?

Current is the rate of flow of charge (Coulombs per second). If more charge flowed out of the narrow end than entered the wide end, charge would be created out of nowhere, violating the fundamental law of conservation of charge.

Q: Doesn't a narrower wire have more resistance, so shouldn't the current drop?

The total resistance of the entire wire determines the *overall* current for a given applied voltage. But once that steady current is established, it must be the exact same value at every single point along the wire, just like water flowing in a pipe.

Q: Why do electrons speed up in the narrow section?

To maintain the same number of electrons passing a point per second (constant current) through a much smaller area, they must travel faster. It's exactly like a wide, slow river entering a narrow gorge; the water has to speed up to get through.

Q: How does the electric field change inside the conductor?

The electric field is the force that pushes the electrons. In the narrow section, they need to move faster, so a stronger electric field is required. Mathematically, $E = J/\sigma = I/(\sigma A)$, so $E$ is inversely proportional to the area.

Q: What does 'steady current' exactly mean?

A steady current means the flow of charge is constant in time. There is no accumulation or depletion of charge at any point in the circuit. It's a state of perfect dynamic equilibrium.

Question

A steady current flows in a metallic conductor of non-uniform cross-section. The quantity/quantities constant along the length of the conductor is/are

Options:

(A) current, electric field and drift speed

(B) drift speed only

(C) current and drift speed

(D) current only

Sigma Percentile · Accepted Answer

### The Steady Flow of Charge

Imagine a river flowing through a canyon. In some places, the river is wide and lazy, and in others, it narrows into roaring rapids. But if the flow is steady, the total volume of water passing any point every second must be exactly the same. If it weren't, water would magically appear or disappear!

The exact same principle applies to electric current. A **steady current** means that the rate of flow of charge is constant over time. Therefore, the current $I$ must be identical at every cross-section of the conductor, regardless of whether the wire is thick or thin. This is a direct consequence of the conservation of charge.

### The Microscopic View: Drift Velocity

Now, let's zoom in and look at the electrons. The relationship between the macroscopic current $I$ and the microscopic drift velocity $v_d$ is given by the famous equation:

$$I = neAv_d$$

Here, $n$ is the number density of free electrons (which depends only on the material), $e$ is the elementary charge, and $A$ is the cross-sectional area.

Since $I$, $n$, and $e$ are all constants in our scenario, we can easily see how drift velocity behaves:

$$v_d \propto \frac{1}{A}$$

This tells us that drift velocity is inversely proportional to the cross-sectional area. Where the conductor is wide, the electrons can afford to drift slowly. But when the conductor narrows, the electrons must speed up to maintain that constant flow of charge—just like the water in the narrow rapids!

### The Driving Force: Electric Field

What causes the electrons to speed up in the narrow sections? It's the electric field! We can understand this using the microscopic form of Ohm's Law:

$$J = \sigma E$$

Where $J$ is the current density ($J = \frac{I}{A}$) and $\sigma$ is the conductivity of the material. Substituting $J$ into the equation gives us:

$$E = \frac{I}{\sigma A}$$

Because $I$ and $\sigma$ are constant, we find that:

$$E \propto \frac{1}{A}$$

The electric field is also inversely proportional to the cross-sectional area. In the narrow regions, the electric field must be stronger to push the electrons faster and maintain the steady current.

### The Final Verdict

By analyzing the physics, we've discovered a beautiful interplay of variables. While the drift speed and the electric field constantly adjust themselves to the changing geometry of the wire, the **current** remains the unwavering constant throughout the entire journey.

Animated Solution for Physics - Current Electricity: A steady current flows in a metallic conductor of non-uniform cross-section. The quantity/quantities constant along the length of the conductor is/are

Select Answer:

Visualized Solution

$I = constant$

$v_{d} = \frac{I}{n e A}$

$v_{d} \propto \frac{1}{A}$

$E = \frac{I}{σ A}$

\text{Conclusion}

\text{The Way Forward}

The Sigma Insight: Ohm's Law, Resistance and Electrical Power

The Steady Flow of Charge

The Microscopic View

The Driving Force

The Final Verdict

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