A long copper strip is 3.0 cm wide and thick. When it carries a steady 42-A current in a 0.80-T magnetic field it produces a 6.5-μV Hall emf. Determine:
(a) the Hall field in the conductor;
(b) the drift speed of the conduction electrons;
(c) the density of free electrons in the metal.

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To determine the Hall field (E_H), use the relationship between the Hall emf (V_H) and the width of the conductor (w). The Hall field is given by the formula:

E H = \frac{V_{H}}{w}

. Substitute the given values for V_H (6.5 μV) and w (3.0 cm) to calculate E_H.

To find the drift speed (v_d) of the conduction electrons, use the Hall effect formula:

E_{H} = v_{d} B

, where B is the magnetic field. Rearrange the formula to solve for v_d:

v_{d} = \frac{E_{H}}{B}

. Substitute the values for E_H (from step 1) and B (0.80 T) to calculate v_d.

To calculate the density of free electrons (n), use the relationship between current (I), drift speed (v_d), cross-sectional area (A), charge of an electron (e), and electron density (n):

I = n e v_{d} A

. Rearrange the formula to solve for n:

n = \frac{I}{e}

. Calculate the cross-sectional area A using the width (w) and thickness (t) of the strip:

A = w t

. Substitute the given values for I (42 A), e (1.6 × 10⁻¹⁹ C), v_d (from step 2), and A (calculated here) to find n.

Ensure all units are consistent before performing calculations. Convert the width (w) and thickness (t) from cm to meters, and the Hall emf (V_H) from μV to volts. This ensures that the results are in standard SI units.

After substituting all the values into the formulas and performing the calculations, you will have the Hall field (E_H), the drift speed (v_d), and the density of free electrons (n). These are the required quantities for parts (a), (b), and (c) of the problem.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Hall Effect

The Hall Effect occurs when a current-carrying conductor is placed in a magnetic field, resulting in a voltage (Hall emf) perpendicular to both the current and the magnetic field. This phenomenon is used to measure magnetic fields and determine charge carrier density in materials. The Hall voltage is directly proportional to the magnetic field strength and the current flowing through the conductor.

Drift Velocity

Drift velocity refers to the average velocity that charge carriers, such as electrons, attain due to an electric field. It is a crucial concept in understanding how current flows in conductors. The drift velocity can be calculated using the relationship between current, charge density, and the area of the conductor, providing insight into the movement of electrons under the influence of an electric field.

Charge Carrier Density

Charge carrier density is the number of charge carriers (like free electrons) per unit volume in a material. It plays a significant role in determining the electrical conductivity of a substance. In metals, this density is typically high, allowing for efficient current flow, and can be calculated using the Hall effect and the known current and magnetic field parameters.