What is meant by "Moving a Test Charge from Infinity"?

Question

This is a section from my physics textbook explaining electric potential energy and electric potential. I am not understand the definition of $\Delta U$, the change in electric potential energy and $V$ the electric potential. I understand that if a charge is placed inside the uniform electric field in the bottom left it will move up or down depending on the charge (Assuming no external force is applied, only electric force) then work is done since a force has been applied over a distance. As the textbook states $\Delta U$ is the change in electric potential energy between a charge at infinity and a test charge at a particular point. Firstly, what is meant by infinity? I'm assuming distance between a test charge and a source charge creating an electric field but the book does not state what infinity is. If infinity is defined as an infinitely large distance, why is $\Delta U$ defined this way? Why couldn't it be "The work required to move an eletric charge from one point to another point against or with the direction of the eletric field"? Why must the first point be infinetly far from the source charge?

In regards to electric potential, the textbook defines the eletric potential at "infinity" to be zero. Looking at the diagram on the left however, how can the lower plate have zero electric potential and not be seperated by an infinite distance from the top plate which is creating the electric field? If I'm not mistaken you can chose any point to be zero electric potential energy just like you can chose any point to be zero gravitational potential energy and therefore the lower plate can be defined as $U=0$ and therefore $V=0$. However, it was explicity stated that $V=0$ if a test charge is an infinite distance from the source charge creating the electric field.

Answer 1

Infinity is a concept in mathematics, and in this context it means that the test charge is originally at infinite distance from your system. Intuitively you would expect that when you are infinitely far from the system, the system cannot affect you. In the case of electric forces, it means the system exerts no force on you. This is intuitively clear, e.g. the Sun is not at an infinitely far distance, but if you jump on the surface of the Earth you can safely ignore the gravitational force from the Sun simply because it's so far away.

To get your local value of $V$, you need to define two points, since we only the difference in potential, $\Delta V$, affects the physics. By convention, we set $V$ at infinity to be $0$. That's why you're seeing the infinity here. It means that if you get a negative number then you know your system has a net negative charge, and a positive number implies the opposite.

"The work required to move an electric charge from one point to another point against or with the direction of the electric field" is sufficient to define $\Delta V$, but the book is actually defining $V$. To set an absolute value on the potential you need something more - you need to define $V$ at a specific point. The book is defining $V_{\infty} = 0$, as per convention.

As for the figure on the left, there is actually a second convention on where to set $V = 0$! The second convention is to set the potential of the Earth to be zero. If this is confusing to you, maybe thinking about kinematics might be more familiar. If you are asked for the potential energy of an object above the ground, you could calculate it using $V = -\frac{G Mm}{r}$, or you could use $PE = mgh$. The first one sets the potential energy at infinity to be $0$, while the second sets the potential energy at the surface of the Earth to be $0$. As you write you can choose any point to be zero potential energy, so both are equally valid. It's just that one or the other might be more convenient for your problem.