Show that \(1 \mathrm{V} / \mathrm{m}\) is the same as \(1 \mathrm{N} / \mathrm{C}\)

The concept of an electric field is fundamental in understanding how electric charges exert force on each other. An electric field is a region of space around a charged particle that experiences a force on another charged particle within that field. The electric field E is defined mathematically as the force F experienced by a positive test charge q divided by the magnitude of the charge itself:

ewline ewline ewline \( E = \frac{F}{q} \).

In this equation, the electric field is directly proportional to the force and inversely proportional to the charge. The units of electric field are newtons per coulomb (N/C), which can also be expressed as volts per meter (V/m). This is because the volt, a unit of electric potential difference, is fundamentally related to the electric field. Understanding the electric field requires recognizing that it describes the force per unit charge, and its direction is the direction in which a positive test charge would move.

In the realm of electricity and electronics, the volt is a unit of measurement that is critical to understanding potential difference and electromotive force. The definition of a volt is based on the work done per unit charge to move the charge between two points. It can be defined as:

ewline ewline ewline \( 1 \text{volt} = \frac{1 \text{joule}}{1 \text{coulomb}} \).

In other words, if one joule of work is required to move a charge of one coulomb across an electric potential difference of one volt, then the voltage between those two points is 1V. This concept is closely linked to the idea of electric field because voltage can be thought of as the electric potential energy per unit charge which is, in turn, related to the concept of an electric field as a force acting over a distance.

Newton's second law of motion is a cornerstone of classical mechanics and it sheds light on the behavior of objects when forces are applied to them. It states that the force applied to an object is equal to the mass of the object multiplied by its acceleration (F = ma). This is often formulated as:

ewline ewline ewline \( F = m \times a \).

Where F is force measured in newtons (N), m is mass in kilograms (kg), and a is acceleration in meters per second squared (m/s²). The law implies that the greater the mass of an object, the more force is needed to accelerate it, and the relation of force, mass, and acceleration is linear. This fundamental law extends its relevance to the field of electromagnetism as well, where the electric forces exerted on charges result in their acceleration and can be related back to the electric field intensity.