Hall Effect
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Direct answer
Drive a current along a conducting slab and pass a magnetic field through its thickness; the charge carriers deflect to one edge, and the resulting transverse voltage — the Hall voltage — builds until its electric field exactly cancels the magnetic force. In equilibrium qE_H = qv_dB, so the Hall voltage is VH = IB/(n e t), where n is the carrier density and t the thickness along the field. The polarity of this voltage reveals whether conduction is by negative electrons or positive holes, the measurement no ordinary resistance experiment can make. The Hall coefficient R_H = 1/(ne) extracts the carrier density directly, and the same balance makes a calibrated slab a Hall probe for measuring magnetic fields.
What you must remember
- Equilibrium condition: the transverse electric field balances the magnetic Lorentz force, E_H = v_d B, with drift velocity v_d = I/(n e A).
- Hall voltage: VH = IB/(n e t), with t the dimension parallel to B; thinner slabs give bigger signals — the thickness that matters is along the field, not along the current or the width.
- Hall coefficient: R_H = 1/(ne); measuring R_H hands you the carrier concentration n, and its sign hands you the carrier type.
- Carrier-sign logic: for a fixed conventional current direction, electrons and holes deflect to the same side (opposite charge, opposite drift direction), so the two cases give opposite Hall voltage — the definitive electron-versus-hole test.
- Magnitude comparison: metals with n near 10²⁹ per cubic metre give microvolt Hall signals; doped semiconductors with n near 10²¹-10²³ give millivolts to volts — Hall devices are built from semiconductors for this reason.
- Hall probe: with n and geometry calibrated, VH measures B linearly; the response is instantaneous and the probe perturbs the field negligibly.
- Sign convention detail: for electron conduction the Hall coefficient comes out negative in the standard geometry, which is why measured R_H signs historically confirmed carriers in metals are negative.
Reading a slab like an instrument
Take a semiconductor strip carrying I = 1 mA along its length, with B = 0.5 T through its thickness t = 0.1 mm and n = 10²¹ carriers per cubic metre. The Hall voltage is VH = IB/(net) = (10⁻³ × 0.5)/(10²¹ × 1.6 × 10⁻¹⁹ × 10⁻⁴) = 0.5/(16) ≈ 0.031 V — about 31 mV, comfortably measurable. Run the same numbers for copper with n = 8.5 × 10²⁹ and the voltage collapses to tens of nanovolts, buried in noise. One line of algebra explains an entire industry: Hall sensors, current clamps that never touch the conductor, and the keyboard-era Hall-effect switches are all this equation with a convenient n.
The sign logic deserves its own pass because it is where marks are won. Send conventional current to the right in a field pointing into the page. If the carriers are positive, they move right with the current, feel qv×B downward, and pile onto the bottom edge — bottom edge positive. If the carriers are electrons, they move left; now a negative charge moving left in the same field again feels a downward force (two sign flips), so electrons also pile on the bottom — but this time the bottom edge goes negative. Same force direction, opposite voltage polarity: measuring the polarity tells you what is actually moving. In p-type and n-type semiconductors this is routine diagnosis, and in metals it is how the electron was confirmed as the carrier.
Where the exam probes
JEE Main keeps to the formula and the drift balance: given I, B, n, t, find VH, or find the drift velocity from a measured voltage. JEE Advanced likes the geometry trap — the formula contains t (along B), but options tempt with the strip width w, since the textbook derivation writes VH = v_d B w and only the substitution v_d = I/(new) removes it; both routes agree only if you track which dimension is which. The second Advanced angle is comparative reasoning: why Hall voltage is larger in semiconductors, what happens to R_H as temperature changes carrier density in doped silicon, and why the effect cannot reveal carrier sign in a perfectly compensated material where electrons and holes both conduct. Assertion–reason formats test the equilibrium idea itself: the transverse voltage stops growing the moment electric and magnetic forces balance, not when "all carriers reach the edge".
Frequently asked questions
What does the sign of the Hall voltage tell you?
Whether carriers are positive or negative, because electrons and holes driven by the same current in the same field deflect to the same edge but leave opposite polarity there.
What is the expression for Hall voltage?
VH = IB/(net), from E_H = v_d B with v_d = I/(neA); the dimension t is the slab thickness along the magnetic field.
Why are semiconductors preferred for Hall devices?
Their carrier density is orders of magnitude smaller than a metal's, so VH — inversely proportional to n — lands in the easily measured millivolt range.
What is the Hall coefficient and what does it measure?
R_H = 1/(ne); its magnitude gives the carrier concentration and its sign gives the carrier type, making it a two-in-one diagnostic.
When does the Hall voltage stop growing?
When the transverse electric field it creates balances the magnetic force per unit charge (E_H = v_dB), after which carriers drift straight through with no net transverse deflection.