a. How much energy in joules is required to move a charge of \(12 \mu \mathrm{C}\) through a difference in potential of \(6 \mathrm{~V}\) ? b. For part (a), find the energy in electron-volts.

In physics, the relationship between charge and energy is foundational. When we talk about 'charge', we're referring to an amount of electricity, which is typically measured in Coulombs (C). Energy, on the other hand, is the capacity to do work, such as moving a charged particle through a wire, and it is measured in joules (J).

To calculate the energy associated with moving a charge, we use the formula:
\[ W = qV \]
where \(W\) is the energy in joules, \(q\) is the charge in Coulombs, and \(V\) is the potential difference in volts. For example, if you have a charge of 12 microCoulombs and a potential difference of 6 volts, the energy required to move that charge is simply the product of these two values, which would be 72 microjoules. This calculation is essential in fields like electronics and electromagnetism where the movement of charges is the basic operation of circuits and electrical devices.

Potential difference, also known as voltage, is a measure of the work done per unit charge to move a charge from one point to another in an electric field. It is a critical concept in circuit theory and electromagnetism, as it drives the flow of electrons through conductors. It is measured in volts (V).

The potential difference is the 'push' that moves charge around a circuit, and this movement of charge is what we observe as an electric current. Without a potential difference, charges do not move, and no current flows. The formula mentioned earlier \(W = qV\) illustrates how potential difference (\(V\)) directly relates to the energy (\(W\)) required to move a charge (\(q\)). In our example, the 6 volts of potential difference dictate how much energy – 72 microjoules – is needed to move the 12 microCoulomb charge.

The electron-volt (eV) is a unit of energy that is commonly used in the field of particle physics and quantum mechanics. It is defined as the amount of kinetic energy gained or lost by a single electron moving through an electric potential difference of one volt.

To convert energy in joules to electron-volts, you use the conversion factor, where 1 eV equals \(1.6 \times 10^{-19}\) joules. For our textbook example, the energy calculated in joules (72 microjoules) can be converted into electron-volts by dividing it by \(1.6 \times 10^{-19}\), yielding an energy of about \(4.5 \times 10^{14}\) eV. This conversion is crucial when dealing with subatomic particles where electron-volts provide a more convenient measure of energy due to the scale and context of the phenomena.