Show that the rate of production of heat per unit volume at a point in a conductor is \(\mathbf{J} \cdot \mathbf{E}\) or \(\sigma \mathbb{E}^{2}\), where \(\mathrm{J}\) is the current density, \(\mathrm{E}\) the electric field and \(\sigma\) the conductivity.

Imagine you have a river, and you want to measure how much water is flowing through a particular section. In electricity, current density serves a similar purpose. It measures how much electric current flows through a unit area of material. Think of it like how many electrons are 'squeezing' through a wire's cross-section. Technically, it's represented by the symbol \textbf{J} and its formula is \( J = \frac{I}{A} \), where \( I \) is the electric current and \( A \) is the cross-sectional area through which the current flows. High current density means a lot of charge is moving through a small area, which can lead to more heat produced in the conductor.

This concept is central to understanding how electrical devices work and for ensuring they operate safely. For instance, wires in your home have to have the right thickness, or cross-sectional area, to safely handle the expected current density without overheating.

An electric field is like an invisible force field that surrounds any electric charge. If you place another charge in this field, it'll feel a force pushing or pulling on it. We describe this field with the symbol \textbf{E}. Now, you might wonder how this field relates to current density. Here's the connection: an electric field can 'push' charges to move, which creates an electric current. The stronger the field, the harder the push, just like how a steeper hill can make you slide down faster. Mathematically, we represent this force per charge as \( E = \frac{F}{q} \), where \( F \) is the force experienced by a charge \( q \).

The presence of an electric field in a material encourages the flow of electricity. It's used in all sorts of technology, from capacitors storing energy to the functioning of touchscreens on your phone.

Think of electrical conductivity as a measure of how good a material is at conducting electricity—it's like how wide or congested a highway is for electron traffic. If conductivity, denoted as \( \sigma \), is high, the material lets electrons pass through easily—imagine a wide-open freeway. But if it's low, electrons face resistance, akin to a jam-packed road during rush hour. This property depends on the material's atomic structure and temperature.

To get technical, conductivity is the inverse of resistivity, which is represented as \( R \). So, \( \sigma = \frac{1}{R} \). Materials like copper and silver have high electrical conductivity, which is why they're used in wires and cables. In contrast, rubber and glass have low conductivity, making them great insulators to prevent unwanted electric currents.

Ohm's Law is like a rulebook for how electric current behaves in a circuit. This law makes it super simple to understand relationships between current, voltage, and resistance. Named after Georg Simon Ohm, it tells us that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance. The oh-so-famous formula goes like this: \( V = IR \), where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance. The cool part? Ohm's Law helps us predict one value if we know the other two, making it a vital tool for designing and understanding electric circuits.

Understanding Ohm's Law is like knowing how to read a map before starting a road trip. It ensures that your electronic devices have the right components to work properly without overheating or failing. For example, you wouldn't use a thin wire for a high-power appliance; it might not handle the current due to its high resistance, based on Ohm's Law.