What's the potential difference between the terminals of a battery that can impart \(7.2 \times 10^{-19} \mathrm{J}\) to each electron that moves between the terminals?

Electrical energy is fundamentally the energy that is caused by moving electric charges, which most commonly are electrons. The more electric charge that flows, and the higher the voltage it's moving across, the more electrical energy is transported.

One can think of electrical energy as the currency of electronics. It powers everything from small devices like watches to large appliances such as refrigerators and massive machines like trains. Electrical energy is measured in joules (J), which quantifies the work done by electricity. In the context of our exercise, each electron is imparted with an energy of \(7.2 \times 10^{-19} \mathrm{J}\). This small amount of energy may seem insignificant, but when considering the billions of electrons flowing in a circuit, the cumulative amount of electrical energy can be substantial.

Analogies to Aid Understanding

It's sometimes helpful to compare electrical energy to water flowing through a pipe. Just like hydraulic energy depends on the water flow and pressure (the water analogue to voltage), electrical energy depends on the flow of electrons and the electrical pressure, or voltage, they're subjected to.

The electron charge is a fundamental constant in physics and one of the most essential aspects of electricity. It is the charge carried by a single electron, which is a tiny particle within atoms that plays a critical role in chemical reactions and electricity.

The charge of an electron is crucial for understanding electrical phenomena because it helps determine how much force an electron will experience in an electric field, and hence how it will move. The standard unit of charge is the coulomb (C), and the charge of one electron is approximately \(-1.6 \times 10^{-19}\) coulombs. This negative sign indicates that the electron has a negative electric charge. When these electrons move through a conductor, like a wire, they create an electric current—a flow of electric charge.

In our exercise, knowing the charge of the electron (\(1.6 \times 10^{-19} \mathrm{C}\) is key to understanding how to calculate the potential difference or voltage that can be exerted on the electron.

Voltage, also known as electric potential difference, is the force that drives an electric current between two points in a circuit. It's comparable to the pressure that pushes water through a pipe. The greater the voltage, the greater the potential energy of the electron charges and the more capable they are of doing work when they reach their destination.

The formula for calculating voltage is straightforward: divide the energy (in joules) by the charge (in coulombs). Symbolically, it is expressed as Voltage (V) = Energy (E) / Charge (Q). Here, \(E\) is the energy transferred to the charged particles measured in joules, and \(Q\) is the charge of the particles measured in coulombs.

In our textbook problem, we used this exact formula. Substituting the values, we have Voltage = \((7.2 \times 10^{-19})/(1.6 \times 10^{-19})\) which simplifies to 4.5V. This means the potential difference between the terminals of the battery is 4.5 volts, which indicates that for every electron that moves from one terminal to another, it's increased in energy by an amount of 4.5 volts.