Electric potential is a scalar quantity; it describes the work that is done per charged particle in order to move it from one point to another. While the electric potential is scalar and does not have direction, the electric charge has a sign that relates to the charge of the particle of interest.
Characteristics of electric potential
Electric potential is positive around an isolated positive charge.
Electric potential is negative around an isolated negative charge.
Electric potential is zero at an infinite distance from the charge under study.
Electric potential in a field
The electric potential of a point charge (q) in a field is proportional to the charge creating the potential, and inversely proportional to the permittivity and distance from the point charge. This is expressed mathematically in the equation below, where V is the electric potential in volts, Q is the point charge, r is the distance measured in metres and εo is the permittivity of a vacuum measured in Farad/metre, equal to 8.85 ⋅ 10-12 F/m.
\[V = \frac{Q}{4 \pi \varepsilon_0 r}\]
From the equation, it can be summarised that for a positive charge q, the electric potential increases when the distance r decreases, as more work will be required to move the positive charge due to the repulsive force. Similarly, for a negative charge q, the distance from the charge decreases, as the positive test charge would move more easily due to the attractive force. This is illustrated below, where the interaction between a positive and a negative charge is shown.
Electric field
To find the potential at a point caused by multiple charges, you have to find the sum of the potential from each charge.
What is electric potential energy?
Electric potential energy is the energy that is needed to move a charge (q) from one point to another in an electric field.
For example, in order to move a positive charge closer to another positive charge, work is needed to overcome the repulsive force. Similarly, when a positive charge is moved away from a negative charge, work is also needed to overcome the attractive force.
The energy transferred to the moving charge is called electric potential energy. The stronger the electric field, the larger the potential energy required to move the charge through the field.
Electric potential energy for two point charges
The electric potential of a pair of point charges is directly proportional to the magnitude of the product of the two charges as shown in the equation below.
Vacuum permittivity ε0 is a constant that represents the tendency of the electric field to permeate in a vacuum. Its value is given below, and is measured in F/m.
\[V [V] = \frac{q_1 \cdot q_2}{4 \pi \varepsilon _0 \cdot r}\]
\[\varepsilon_0 = 8.885 \cdot 10^{-12} F/m\]
Change in electric potential energy can be found by utilising the respective distance of each unit charge.
\[V = \frac{q_1 \cdot q_2}{4 \pi \varepsilon _0 \cdot r} \cdot \Big( \frac{1}{r_1} - \frac{1}{r_2} \Big)\]
Electric potential and work
The electric potential can also be expressed mathematically in terms of work; remember, it is the work required to move a charge through an electric field. The work is equal to the product of the electric potential and the charge causing electric potential. This is shown below, where ΔV is the change in electric potential measured in volts, and Q is the charge measured in Coulombs.
\[W [J] = \Delta V Q\]
What is the electric potential gradient?
The electric potential gradient is the varying electric potential across an electric field. The electric field at any point is equal to the negative gradient of the potential distance at that point. The potential difference is shown below.
The gradient is defined by the equipotential lines which are represented in orange circular dotted lines and show the electric potential in an electric field. These are always perpendicular to the electric field lines which are illustrated with blue lines. The equipotential lines express the strength of the electric potential. The denser the equipotential lines the stronger the potential.
Electric field and equipotential lines
What is electric potential difference?
Electric potential difference is the work needed to move a charged particle in an electric field, from point A to point B. This is expressed in the equation below, where E is the electric field strength, V is the electric potential in volts, and r is the distance between the two points of interest in metres.
The negative sign shown below expresses the direction of the electric field. It is always outwards from the positive charge and inwards towards the negative charge (take a look at the first image).
\[E [N/C] = -\frac{\Delta V}{\Delta r}\]
An electric generator with a spherical shape has a radius of 10cm and generates a potential of 150kV. Find the electric potential at a 25cm distance from the generator.
Solution:
We begin by finding the charge using the potential equation and re-arranging to make Q the subject, then substituting the given values.
\[V[V] = \frac{Q}{4 \pi \varepsilon _0 r} \qquad Q = V[V] 4 \pi \varepsilon _0 [F/m]r[m] = 150 \cdot 1000 \cdot \pi \cdot 4 \cdot 8.85 \cdot 10^{-12} \cdot 0.1 = 1.67 \cdot 10^{-6} [C]\]
Then we proceed to find potential by using the potential equation. But now we use the total distance at 25cm away from the generator, which is the distance of the radius plus the distance from the generator.
\[R = \text{radius of sphere + distance from generator} \quad V = \frac{Q}{4 \pi \varepsilon_0 r} = \frac{1.67 \cdot 10^{-6}C}{4 \cdot \pi \cdot 8.85 \cdot 10^{-12} F/m \cdot (0.25 + 0.1)m} = 42. 857 kV\]
Electric Potential - Key takeaways
Electric potential is the work required to move a charge from one point to another.
The electric potential difference is the change in electric field strength between two points of an electric field.
The electric potential (V) decreases, as the distance between the point under study and the electric potential source increases.
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