Beatrice Lang
Electric potential, also known as voltage, is the electric potential energy per unit charge. It is a scalar quantity with no direction and is measured in joules (J) and coulombs (C). The electric potential at a point P due to a charge q is inversely proportional to the distance between them. The electric potential energy of an object depends on its charge and its relative position to other electrically charged objects. The electric potential difference between two points, A and B, is defined as the work done to move a positive unit charge from A to B. The change in potential energy of a charge q when moved from point A to point B is the work done by the external force Fext, which is equal to the change in the electrostatic potential energy of the particle in the external field. The electric potential at any point can be defined as the amount of work done to move a test charge from infinity to that point. The electric potential of a point charge is given by V = kq/r, where k is a constant.
| Characteristics | Values |
|---|---|
| Electric potential formula | V = kq/r |
| Electric potential energy formula | Up = -q∫rp E·dr |
| Electric potential energy definition | Work done by an external agent in bringing the charge from infinity to the given point |
| Electric potential energy scalar quantity | V |
| Electric potential energy SI unit | Joule (J) |
| Electric potential | Work done in moving a test charge from infinity to that point |
| Electric potential difference between two points | Work done to move a positive unit charge from point A to point B |
| Electric potential SI unit | J/C or volt (V) |
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What You'll Learn
Electric potential energy
The electric potential at any point is the amount of work done in moving a test charge from infinity to that point. The electric potential energy of a test charge is given by the formula:
\[U_p = q_tV_p = q_tk\sum_1^N \frac{q_i}{r_i}\]
Where \(U_p\) is the electric potential energy, \(q_t\) is the test charge, \(V_p\) is the electric potential, \(k\) is a constant, \(q_i\) is the charge at point \(P\) in space, and \(r_i\) is the distance from the charge at point \(P\).
The electric potential due to a point charge is given by:
\[V = \frac{kq}{r}\]
Where \(V\) is the electric potential, \(k\) is a constant, \(q\) is the charge, and \(r\) is the distance from the charge.
The change in potential energy of a charge \(q\) when being moved from point A to point B is the work done by the external force \(F_{ext}\) in moving the charge. The change in potential energy is given by:
\[Δ U = q Δ V\]
Where \(\Delta U\) is the change in potential energy, \(q\) is the charge, and \(\Delta V\) is the change in potential.
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Electric potential at infinity
Electric potential, also known as voltage, is the amount of electric potential energy per unit charge at a point in an electric field. It is calculated as the electric potential energy at a point divided by the amount of charge at that point.
The electric potential at a point due to a point charge can be calculated using the formula:
\[ V = \frac{kq}{r}\]
Where $k$ is a constant, $q$ is the point charge, and $r$ is the distance from the point charge.
The electric potential at infinity is defined as zero. This is a mathematical convenience, as when calculating physical effects, only the differences in potential matter. The potential at a finite distance $r$ from a point charge $q$ is given by $V = \frac{kq}{r}$, and as $r$ approaches infinity, the potential $V$ approaches zero.
The electric potential energy of a test charge $q$ at a point $P$ in space due to $N$ charges fixed at distances $r_1, r_2, ..., r_N$ is given by:
\[ V_p = k\sum_{i=1}^N \frac{q_i}{r_i}\]
By substituting the values of $V_p$ and $q_i$ in the equation $U_p = q_tV_p$, we can calculate the electric potential energy of the test charge.
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Electric potential difference
Electric potential is a scalar quantity, meaning it has magnitude but no direction. It is defined as the electric potential energy per unit charge. The SI unit of electric potential is, therefore, volt (V), derived from the SI units of electric potential energy (joule, J) and charge (coulomb, C).
The electric potential difference between two points, A and B, is defined as the work done to move a positive unit charge from point A to point B. This is calculated using the formula:
ΔV = VA − VB
Where ΔV is the change in electric potential, and VA and VB are the electric potentials at points A and B, respectively.
The electric potential at a point P due to a charge q is inversely proportional to the distance between them. This relationship is described by the equation:
V = kq/r
Where k is a constant equal to 8.99 x 10^9 Nm^2/C^2, q is the charge, and r is the distance from the charge.
The electric potential energy of a charge at a point in space depends on its distance from other charges. If a charge q_i at point P in space has distances of r_1, r_2, ..., r_N from N charges fixed in space, the electric potential at point P is given by:
V_p = kΣ_1^N q_i/r_i
Where V_p is the electric potential at point P, k is a constant, q_i is the individual charge, and r_i is the distance from each charge.
By substituting the electric field formula (E = F/q) into the equation for potential energy (U_p), we can derive an alternative formula for electric potential:
V_p = −∫_R^P E∙d l
Where V_p is the electric potential at point P, E is the electric field, and d l is an infinitesimally small path element.
The electric potential and electric potential energy of a system of charges can be calculated using the superposition principle. This principle states that the total electric potential (or electric potential energy) of a system of charges is equal to the sum of the electric potentials (or electric potential energies) of each individual charge.
The electric potential and potential energy of a charge are closely related to the work done by an external force. The work done by an external force in moving a charge from one point to another is equal to the change in the electric potential energy of the charge. This relationship is described by the equation:
ΔU = qΔV
Where ΔU is the change in electric potential energy, q is the charge, and ΔV is the change in electric potential.
In summary, electric potential difference is a fundamental concept in physics that describes the work required to move a positive unit charge between two points. It is a scalar quantity with units of volts (V). The electric potential at a point is influenced by nearby charges and is calculated using equations that account for the magnitudes of charges and their distances. Understanding electric potential and potential difference is crucial for analyzing electrical circuits, electric fields, and the behaviour of charged particles.
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Work done by external force
Electric potential is the work done per unit charge in bringing a small test charge from infinity to a position r. The electric potential due to a point charge is given by:
> V = kq/r
Where k is a constant equal to 8.99 x 10^9 Nm^2/C^2. The electric potential energy of the test charge is:
> Up = qtVp
The work done by external forces in electric fields is determined by the direction in which the charge is moved. If a charge is moved against the direction of the electric field, the work done is positive and the potential energy of the charge increases. On the other hand, if the charge is moved in the direction of the electric field, the work done is negative, and the potential energy of the charge decreases.
For example, consider a positive point charge q at the origin. To calculate the potential caused by q at a distance r from the origin relative to a reference of 0 at infinity, let P = r and R = infinity, with dl = dr = rdr and E = kq/r^2.r. The work done by the external force to bring a positive charge from infinity to a point P close to the origin is W, and the work done by the field is -W. The change in potential energy is the same in both cases, but in the case of work done by the field, the work done is stored as electrostatic potential energy, while in the case of work done by the external force, the work done is positive, and the energy is taken by some external source.
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Electric potential energy formula
Electric potential energy is a type of potential energy that results from conservative Coulomb forces. It is associated with the configuration of a particular set of point charges within a defined system. An object may be said to have electric potential energy due to its own electric charge or its relative position to other electrically charged objects. The term "electric potential energy" is used for systems with time-variant electric fields, while "electrostatic potential energy" is used for systems with time-invariant electric fields.
The electric potential energy of a system of point charges is defined as the work required to assemble the system by bringing the charges close together. It is measured in joules and can be calculated using a formula similar to Coulomb's law:
> U = k * (q1 * q2) / r
Where k is the Coulomb's law constant, q1 and q2 are the two interacting charges, and r is the distance between them.
The electric potential energy of a charge in an electric field is independent of the path chosen. The formula for the electric potential energy of a test charge q_i at a point P in space with distances of r1, r2, ..., rN from the N charges fixed in space is:
> U_p = q_t * V_p = q_t * k * Σ(q_i / r_i)
Where V_p is the potential at point P, and q_t is the test charge.
The electric potential due to a point charge q at a distance r can be calculated as:
> V = k * q / r
Where k is a constant equal to 8.99 x 10^9 N * m^2 / C^2.
The electric potential energy of a system of two charges is equal to the electrostatic potential energy of a charge in the electrostatic potential generated by the other. The electrostatic potential energy of a system of three charges can be calculated using the following formula:
> U_E = (1 / 4πε_0) * [Q1*Q2 / r12 + Q1*Q3 / r13 + Q2*Q3 / r23]
Where ε_0 is the vacuum permittivity, and r12, r13, and r23 are the distances between the charges.
The electrostatic potential energy of one point charge q at position r in the presence of an electric field E is defined as the negative of the work W done by the electrostatic force to bring it from the reference position rref to that position r:
> U_E = -W_rref→r = -∫_rref^r q*E * dr
Where E is the electrostatic field, and dr is the displacement vector from the reference position to the final position.
The change in potential energy ΔU as two charges are brought closer or moved farther apart depends on the relative types of charges. If the charges are of the same type (both positive or both negative), the electric potential energy is positive. If the charges are of opposite types, the electric potential energy is negative.
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Frequently asked questions
The electric potential at a point P due to a charge q is inversely proportional to the distance between them. The electric potential at any point is defined as the amount of work done in moving a test charge from infinity to that point. The formula for the electric potential \(V\) of a point charge is given by:
\[ V = \frac{kq}{r} \]
where \(k\) is a constant equal to \(8.99 \times 10^9 \, \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2\).
The electric potential difference between two points A and B is defined as the work done to move a positive unit charge from A to B. The SI unit of potential difference is the volt (V). The formula for the change in potential energy of a charge q when being moved from point A to point B is:
\[ \Delta V = V_B - V_A \]
The electric potential energy of a test charge \(q_i\) at point P in space with distances of \(r_1, r_2, ... r_N\) from the \(N\) charges fixed in space is given by:
\[ V_p = k\sum_{1}^{N} \frac{q_i}{r_i} \]
Therefore, the electric potential energy of the test charge is:
\[ U_p = q_tV_p = q_tk\sum_{1}^{N} \frac{q_i}{r_i} \]
