Hall Effect Voltage Calculator

Overview: What This Hall Effect Calculator Does

This calculator lets you solve the Hall effect relation for any one of its key variables: Hall voltage V_H, current I, magnetic field B, carrier density n, sample thickness t, or charge magnitude q. It is based on the ideal Hall effect formula

Hall voltage relation:

Given any five of these quantities, the tool computes the sixth using this relationship. It is intended for physics and electronics students, laboratory users, and engineers who need a quick way to estimate Hall voltages or related parameters under idealized conditions.

How the Hall Effect Works

The Hall effect appears when an electric current flows through a conductor or semiconductor in the presence of a magnetic field that is perpendicular to the current. Moving charge carriers (electrons or holes) experience a magnetic force that pushes them sideways. As they accumulate on one side of the sample, an electric field builds up across the sample’s thickness. The resulting potential difference is called the Hall voltage, V_H.

In a typical Hall bar: current flows along the length, the magnetic field points perpendicular to the sample surface, and the Hall voltage is measured across the width or thickness. The sign of this voltage reveals the sign of the dominant charge carriers, and the magnitude tells you about the carrier density and the strength of the magnetic field.

Hall Voltage Formula and Symbols

The ideal Hall effect relation used by the calculator can be written as

where:

• V_H is the Hall voltage in volts (V).
• I is the current through the sample in amperes (A).
• B is the magnetic flux density in tesla (T).
• n is the charge carrier density in m&supmin;3 to the power of minus three (m⁻³).
• q is the magnitude of charge of each carrier in coulombs (C). For electrons this is approximately 1.602 × 10⁻¹⁹ C.
• t is the sample thickness (the dimension across which the Hall voltage is measured) in meters (m).

Rearranging the formula allows you to solve for any variable. For example:

• Solve for current: I = VH n q t / B
• Solve for magnetic field: B = I / (n q t) × 1 / VH
• Solve for carrier density: n = I B / (q t VH)
• Solve for thickness: t = I B / (n q VH)

The calculator performs these algebraic rearrangements internally. You choose which variable to solve for, enter the others, and the corresponding expression is evaluated numerically.

Using the Calculator and Interpreting Results

To use the tool, select which quantity you want to compute (for example, Hall voltage V_H), then enter known values for the remaining variables. All inputs are in SI units:

• Current I: amperes (A).
• Magnetic field B: tesla (T).
• Carrier density n: m⁻³.
• Thickness t: meters (m).
• Charge magnitude q: coulombs (C); by default the elementary charge.

After calculation, the result tells you how strong the Hall response is under the specified conditions. A larger Hall voltage means a stronger sideways electric field has developed to balance the magnetic deflection of the charges. Thin samples, low carrier density, strong magnetic fields, and high currents all tend to increase the Hall voltage.

In practice, measured Hall voltages are often in the millivolt or microvolt range. If your inputs lead to unreasonably large voltages (for example, many volts for modest lab conditions), double-check units and orders of magnitude.

Worked Example

Suppose you have a doped silicon sample 1 mm thick, carrying a current of 20 mA in a magnetic field of 0.3 T. Assume an electron carrier density of 1 × 10²¹ m⁻³ and use the elementary charge q = 1.602 × 10⁻¹⁹ C. You want to find the Hall voltage.

1. Convert thickness to meters: t = 1 mm = 1 × 10⁻³ m.
2. Write down known values: I = 0.02 A, B = 0.3 T, n = 1 × 10²¹ m⁻³, q = 1.602 × 10⁻¹⁹ C, t = 1 × 10⁻³ m.
3. Use the Hall formula: VH = I B / (n q t).
4. Compute the denominator: n q t = (1 × 10²¹) × (1.602 × 10⁻¹⁹) × (1 × 10⁻³).
5. First multiply n and q: (1 × 10²¹) × (1.602 × 10⁻¹⁹) = 1.602 × 10².
6. Now include t: 1.602 × 10² × 10⁻³ = 1.602 × 10⁻¹ = 0.1602.
7. Compute the numerator: I B = 0.02 × 0.3 = 0.006.
8. Divide: V_H = 0.006 / 0.1602 ≈ 0.0374 V = 37.4 mV.

The calculator will return approximately 3.74 × 10⁻² V, or 37.4 mV. This voltage is small but measurable with standard instrumentation. If you double the sample thickness to 2 mm while keeping everything else fixed, the denominator doubles and the Hall voltage halves. Similarly, if you reduce carrier density by a factor of 10 (for example, using a more lightly doped semiconductor), the Hall voltage increases by a factor of 10.

These trends are useful when designing Hall sensors or planning experiments: thin, low-density semiconductor samples in strong magnetic fields produce the most pronounced Hall voltages, at the cost of higher resistivity and potentially higher measurement noise.

Comparison of Parameter Effects

The table below summarizes how each parameter appears in the Hall voltage expression and how changing it (with all other variables fixed) affects the magnitude of V_H.

Note that the sign of the Hall voltage depends on the sign of the charge carriers and the orientation of the magnetic field and current. The calculator uses the magnitude of the charge |q|; changing the sign of q would simply flip the sign of V_H in a real experiment.

Assumptions and Limitations

The Hall effect relation used here is an idealized model. When you apply the calculator to real materials or devices, keep these assumptions and limitations in mind:

• Uniform magnetic field: The formula assumes that B is uniform across the entire cross section of the sample. In fringe fields or near magnets with strong gradients, the actual Hall voltage may differ.
• Single carrier type: The expression V_H = I B / (n q t) assumes that conduction is dominated by a single type of charge carrier (electrons or holes). Many real materials, especially at intermediate temperatures, have mixed conduction that modifies the effective Hall response.
• Simple geometry: The calculator treats the sample as a rectangular bar with well-defined thickness and current distribution. Thin films with complex geometries, patterned Hall plates, or devices with guard rings require geometry factors or numerical modeling.
• Steady-state current: Transient effects and high-frequency operation are ignored. The model is most accurate for DC or slowly varying currents.
• Linear material response: It is assumed that material properties such as carrier density and mobility do not change with applied field or current in the range considered. Strong heating or very high fields can invalidate this.
• SI units only: All entries and results are in SI. If you work in cm, gauss, or other units, convert to meters and tesla before using the calculator.

Because of these simplifications, measured Hall voltages in real devices may contain calibration factors or systematic offsets. Use the calculator for estimates, cross-checks, and conceptual understanding rather than as a substitute for careful calibration.