Some hypothetical metal is known to have an electrical resistivity of \(3.3 \times 10^{-8}(\Omega-\mathrm{m})\) Through a specimen of this metal \(15 \mathrm{mm}\) thick is passed a current of 25 A; when a magnetic field of 0.95 tesla is simultaneously imposed in a direction perpendicular to that of the current, a Hall voltage of \(-2.4 \times 10^{-7} \mathrm{V}\) is measured. Compute (a) the electron mobility for this metal, and (b) the number of free electrons per cubic meter

To understand how electrical resistivity ties into the Hall effect, picture a metal as a highway where electrons are cars speeding along. Electrical resistivity, symbolized by \rho, denotes the metal’s natural tendency to resist the flow of these 'electron cars'. If the resistivity is high, the metal is like a highway during rush hour – it's difficult for the cars to move quickly due to congestion. Conversely, low resistivity means a clear highway where electrons can cruise at higher speeds.

Electrical resistivity is crucial when we talk about the Hall effect in metals because it gives us a way to measure the inherent opposition to current flow within the material. When a magnetic field is applied perpendicular to the electric current, such as in the Hall effect experiments, it creates a sideways force called the Lorentz force, which deflects electrons and leads to a buildup of charge on one side of the material. This buildup is what we measure as the Hall voltage. The resistivity of the metal can then be used to determine other important properties of the metal, like electron mobility, which was calculated in the step by step solution using the formula \( R = \frac{L \times \rho}{A} \).

Just as we gauge how well a car handles using its ability to quickly and smoothly change speeds or direction, electron mobility (\(\mu\)) tells us how nimble electrons are within a metal. In simpler terms, mobility measures how quickly an electron can move when an electric field is applied. Think of electron mobility as the agility level for electrons – a higher mobility means they can zip around easier, responding more sensitively to the electric 'push'.

When you’re trying to calculate how mobile these electrons are, as in the Hall effect, you’re really getting an idea of how effectively the material will conduct electricity. That’s why the hypothetical metal's electron mobility was calculated using the formula \( \mu = \frac{V_H}{B} \times \frac{1}{R} \) in the step by step solution. Good conductors have high electron mobility, enabling fast and efficient energy transfer, which is especially important in applications like electronics and data transmission.

While learning about electrical concepts, you'll hear about 'free electrons', but what does this actually mean? In layman's terms, free electrons are the ones not bound tightly to atoms and can move freely through the metal, like bees buzzing around a hive freely without being attached to any particular comb. The density of these free electrons per cubic meter (\(n\)) significantly affects a metal's electrical properties.

It's also the reason why some metals are better conductors than others. If a metal has more free electrons available, it can conduct more electricity. This concept is key in the Hall effect, as we saw in the exercise where the number of free electrons per cubic meter was found using the formula \( n = \frac{I \times B}{V_H \times e \times t} \). This value isn't just a number; it tells us about the sea of electrons that are ready and waiting to leap into action and carry a current, making it an essential piece of the electrical puzzle.