Umair Gaines
Electric potential, also known as electric field potential, potential drop, or electrostatic potential, is a fundamental concept in physics that relates to the behaviour of electric fields and charged objects. It is defined as the electric potential energy per unit of electric charge. In simpler terms, it represents the amount of work required to move a test charge from a reference point to a specific point within a static electric field. This reference point is typically Earth or infinity, but it can be any chosen location. The electric field and electric potential are closely related, and understanding their connection is essential for deriving the electric field from the electric potential. This relationship is expressed in vector calculus notation as E = −grad V, where E represents the electric field and V represents the electric potential.
| Characteristics | Values |
|---|---|
| Definition | Electric potential is the work done per unit charge in moving the charge from infinity to that distance |
| Electric potential energy | The electric potential energy of a test charge divided by its charge for every location in space |
| Electric field | The force on a test charge divided by its charge for every location in space |
| Relationship between electric field and electric potential | The electric field is the gradient of the electric potential |
| Electric potential due to a point charge | \(V = − \int\mathbf{E}\cdot\mathrm{d}\mathbf{s} =−\int E\cos\theta \mathrm{d}s\) |
| Electric potential due to a positive stationary charge | \(V = − \int E \cos180° \mathrm{d}s = − KQ_1 \int \cos180°\frac{1}{r^2} \mathrm{d}s\) |
| SI derived unit | Volt (V) |
| Electric potential in Lorenz gauge | A retarded potential that propagates at the speed of light and is the solution to an inhomogeneous wave equation |
| Electric potential in CGS-Gaussian system | Many equations would be altered |
| Electric potential in vector calculus notation | \(E = −\nabla V\) |
| Electric potential in Cartesian coordinates | The del operator is the sum of the partial derivatives in the three unit vector directions |
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What You'll Learn
Electric potential and electric field relationship
Electric potential, also known as electric field potential or electrostatic potential, is a fundamental concept in physics that relates to the behaviour of electric fields and charges within a given system. It is defined as the electric potential energy per unit of electric charge. This definition underscores the relationship between electric potential and the work done to move a charge within an electric field.
The electric field and electric potential are intimately linked. In calculus terms, the electric field can be expressed as the gradient of the electric potential. This relationship holds true in three-dimensional space, where the gradient represents the change in potential across multiple dimensions. The electric field is a vector, possessing both magnitude and direction, and it can be derived from knowledge of the electric potential at every point in a region of space.
The electric potential at a specific point in space is influenced by the presence of charges. For instance, electric field lines point away from positive charges and towards negative charges. The magnitude of the electric field vector is determined by the quantity of the charge. Furthermore, the electric potential due to an idealized point charge is continuous in all space except at the location of the point charge itself.
The relationship between electric potential and electric field can be described by the equation E = −grad V, where E represents the electric field and V represents the electric potential. This equation illustrates that the electric field is the negative gradient of the electric potential. The direction of the electric field is determined by the steepest decrease in potential, moving away from a given point.
In certain scenarios, such as the presence of time-varying magnetic fields, the electric field cannot be solely described by a scalar potential. Instead, it is expressed using both the scalar electric potential and the magnetic vector potential. This combination of potentials forms a four-vector, highlighting the intricate interplay between electric and magnetic fields.
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Electric potential energy
The electric potential at a particular point in space around a point charge can be calculated using Coulomb's Law, which states that the electrostatic potential is directly proportional to the inverse of the distance from the point charge. This relationship is continuous in all space except at the location of the point charge. The electric potential due to a positive charge increases as the distance from the charge decreases, while the opposite is true for a negative charge.
The electric potential energy of a system of charges is determined by the separation between the charges. When two unlike charges, such as an electron and a proton, are brought closer together, the electrostatic potential energy of the system decreases. Conversely, when two like charges, such as two electrons or protons, are brought closer together, the electrostatic potential energy of the system increases.
In vector calculus, the electric field can be derived from the electric potential using the equation E = −grad V, where E represents the electric field and V represents the electric potential. This equation specifies the calculation of the electric field at a given point, taking into account both its direction and magnitude. The direction of the electric field is away from positive charges and towards negative charges, with the magnitude given by the change in potential across a small distance.
The SI unit of electric potential is the volt (V), named after Alessandro Volta, and it represents the electric potential difference between two points in space.
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Electric potential in scalar field
Electric potential, also known as electric field potential, potential drop, or electrostatic potential, is defined as electric potential energy per unit of electric charge. In other words, it is the amount of work needed to move a test charge from a reference point to a specific point in a static electric field.
The electric potential is a scalar field, meaning it is a directionless value that depends only on its location. The electric potential energy of a test charge is divided by its charge for every location in space. The electric field and electric potential are related by displacement. The electric field is the force on a test charge divided by its charge for every location in space. Because it is derived from a force, it is a vector field.
In electrodynamics, when time-varying fields are present, the electric field cannot be expressed solely as a scalar potential. Instead, it is expressed as both the scalar electric potential and the magnetic vector potential, which together form a four-vector. The electric potential is a continuous function in all space, as a spatial derivative of a discontinuous electric potential would result in an electric field of infinite magnitude.
The electric potential due to an idealized point charge is continuous in all space except at the location of the point charge. The electric scalar potential is undefined to an additive constant, and the usual convention is to assume that the potential is zero at infinity. The scalar potential generated by a set of discrete charges can be expressed as a simple volume integral involving the charge distribution.
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Electric potential in vector calculus
Electric potential, also known as electric field potential, potential drop, or electrostatic potential, is a fundamental concept in physics that relates to the behaviour of electric fields and charges. It is defined as the electric potential energy per unit of electric charge. In other words, it represents the amount of work required to move a test charge from a reference point to a specific point within a static electric field.
When considering electric potential in vector calculus, several key concepts come into play. Firstly, the electric field, denoted by a bold uppercase letter E, is a vector quantity. It represents the force per unit charge acting on an imaginary test charge at any location in space. The electric field is derived from the force and is, therefore, a vector field.
The electric potential, on the other hand, is derived from energy and is thus a scalar field. It is denoted by V or occasionally φ (phi). The electric potential at the reference point, typically Earth or a point at infinity, is defined as zero units. The electric potential at any location in a system of point charges is given by the sum of the individual electric potentials due to each point charge in the system.
The relationship between the electric field and electric potential is essential. In calculus terms, the electric field is the gradient of the electric potential. This relationship holds in three-dimensional space, where the gradient is the equivalent of a derivative in higher dimensions. The electric field and electric potential are also related by a path integral that is independent of the path taken due to the conservative nature of electricity.
In the presence of time-varying magnetic fields, the electric field cannot be expressed solely as a scalar potential. Instead, it is described using both the scalar electric potential and the magnetic vector potential, which together form a four-vector. This combination allows for a concise representation of classical electromagnetism, particularly when using the Lorenz gauge and performing Lorentz transformations.
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Electric potential in Cartesian coordinates
Electric potential, also known as electric field potential or potential drop, is defined as electric potential energy per unit of electric charge. It is a scalar quantity denoted by V or φ. The electric potential at the reference point is defined to be zero units, with the reference point typically being the Earth or a point at infinity.
Electric potential can be used to explain the origin of an electric field. The electric field "flows" from regions of high potential to regions of low potential. The electric field and electric potential are related by displacement. The electric field is the force on a test charge divided by its charge for every location in space. It is a vector field. The electric potential, on the other hand, is the electric potential energy of a test charge divided by its charge for every location in space. It is a scalar field.
In Cartesian coordinates, the del operator is the sum of the partial derivatives in the three unit vector directions. The general formula is:
E = -grad V
This can be broken down into the following equations:
E_x = - dV/dx
E_y = - dV/dy
E_z = - dV/dz
The electric potential equation for a dipole in Cartesian coordinates can be expressed as:
V(x,y,z) = q/√(x^2+y^2 +(z-d/2)^2) - q/√(x^2+y^2 +(z+d/2)^2)
Where x and y are Cartesian coordinates, and z is the electric potential.
The electric potential equation for an E-field in Cartesian coordinates can be calculated using the following steps:
- Start with the equation: dV = E·dr
- Integrate term by term to calculate V(x, y, z)
- For example, calculate: -dV/dx = x
V = - 1/2 x^2 + f(y, z)
- Continue integrating the rest of the terms
- Solve for the constant of integration by knowing the value of V at a certain point in space
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Frequently asked questions
Electric potential, also known as electric field potential or electrostatic potential, is defined as electric potential energy per unit of electric charge. It is the amount of work needed to move a test charge from a reference point to a specific point in a static electric field.
The electric field is the derivative of electric potential. In vector calculus notation, the electric field is given by the negative of the gradient of the electric potential, E = −grad V.
Electric potential and electric field are related by displacement. The electric field is the force on a test charge divided by its charge for every location in space. The electric potential is the electric potential energy of a test charge divided by its charge for every location in space.
The SI derived unit of electric potential is the volt, denoted as V.
