Molly Ayala
Electric potential is a scalar field that provides information about the energy landscape produced by an electric field. It is easier to graph compared to an electric field because it is a scalar quantity, meaning direction does not need to be taken into account. Electric potential-distance graphs can be used to determine the potential difference from field-distance graphs. The electric potential around a positive charge decreases with distance and increases with distance around a negative charge. The gradient of the graph at any point is equal to the field strength E at that point.
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What You'll Learn
Electric potential is a scalar field
The electric field can be expressed as both the scalar electric potential and the magnetic vector potential. The electric potential and the magnetic vector potential together form a four-vector, and their behaviour is mixed under Lorentz transformations. The electric potential is considered a component of this four-vector, and it is invariant under certain transformations. For example, the electric potential is a scalar under the rotation group and more generally under the Galilean Group.
The scalar nature of electric potential can be understood through its relationship with force. When a test charge is brought to a certain point, we exert a force that must be equal to or greater than the force exerted by the electric field to overcome it. As a result, the forces cancel each other out, and the direction becomes undetermined. Since vectors are defined by both magnitude and direction, the absence of a direction makes electric potential a scalar quantity.
The concept of scalar electric potential is particularly useful in classical electrostatics. The electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a scalar. This scalar potential simplifies calculations and is commonly used in equations such as Coulomb's law and the Poisson equation. Once the scalar potential is determined, the electric field can be found by taking the negative of its gradient.
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Electric potential vs distance graphs
To understand electric potential vs distance graphs, let's first break down the two key components: electric potential and distance.
Electric Potential:
Electric potential, also known as voltage, is a fundamental concept in physics that describes the amount of electric potential energy per unit charge at a specific point in an electric field. It is a scalar quantity, meaning it has magnitude but no direction. Electric potential is typically measured in volts (V).
Distance:
In the context of electric potential vs distance graphs, distance refers to the separation between two points in an electric field. This could be the distance between two charges, or the distance from a charge to a specific point in space. Distance is typically measured in meters (m).
Understanding the Graph:
Now, let's bring these concepts together and understand how they are represented in an electric potential vs distance graph:
- Graph Axes: The graph will have electric potential on the vertical axis (often labeled as "V") and distance on the horizontal axis (often labeled as "r").
- Positive and Negative Charges: The behaviour of the graph will differ depending on whether you're dealing with a positive or negative charge. For a positive charge, the electric potential decreases as the distance (r) increases, following a 1/r relationship. On the other hand, for a negative charge, the electric potential increases with distance, following a -1/r relationship.
- Gradient and Field Strength: The gradient of the graph at any point is crucial. It represents the electric field strength (E) at that particular point. The gradient is given by the derivative of the electric potential curve with respect to distance.
- Area and Potential Difference: The area under the curve of an E-r graph (field strength vs distance) between two points represents the potential difference (ΔV) between those points. This is a useful way to determine the potential difference due to a charge.
- Point Charges: When dealing with point charges (charges with zero radius), the potential diverges, meaning it becomes infinite as you approach the charge (r = 0). This is because you can get arbitrarily close to a point charge.
- Finite Charges: For charges with a finite size, the electric field inside the charge remains constant, and outside the charge, it follows a similar behaviour to a point charge graph.
By understanding these relationships and behaviours, you can interpret electric potential vs distance graphs for various charge distributions and gain insights into the electric potential energy and field strength at different points in an electric field.
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Electric potential inside a charge
Electric potential, also known as electric field potential or electrostatic potential, is defined as the amount of work or energy needed per unit of electric charge to move a charge from a reference point to a specific point in an electric field. In other words, it is the energy per unit charge for a test charge, which is so small that the disturbance to the field is negligible. The reference point is typically the Earth or a point at infinity, and the electric potential at this point is defined as zero.
The SI unit of electric potential is the volt, denoted as V, and the electric potential difference between two points in space is known as voltage. The electric potential inside a charge can be understood by considering a system of N charges. Each charge in the system produces its own electric potential at a given point, independent of the other charges. The net electric potential at that point is equal to the sum of these individual electric potentials.
For a point charge, such as an electron, the electric potential is continuous in all space except at the location of the point charge. The electric field is not continuous across a surface charge, but it is also not infinite at any point. Therefore, the electric potential is continuous across a surface charge. The electric potential due to a point charge is proportional to 1/r, where r is the distance from the point charge.
For a finite-sized charged particle, the field inside the charged particle is not divergent. For example, in the case of a hollow sphere with some charge, the potential is constant inside the sphere. However, outside the sphere, the potential follows a different behaviour.
The electric potential inside a metal sphere can be calculated using the formula:
V = k * (q/r)
Where V is the electric potential, k is the electrostatic constant, q is the charge, and r is the distance.
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Electric potential and point charges
Electric potential, also known as electric field potential or electrostatic potential, is defined as the amount of work or energy required to move a unit of electric charge from a reference point to a specific point in an electric field. The reference point is typically the Earth or a point at infinity, and the electric potential at the reference point is defined as zero.
In the context of point charges, the electric potential at any location in a system of point charges is equal to the sum of the individual electric potentials due to each point charge in the system. This is because the addition of potential (scalar) fields is much simpler than the addition of electric (vector) fields. The electric potential at a point in an electric field is influenced by the presence of a positive charge, which exerts a force due to the electric field. As a result, the electric potential energy of an object in an electric field depends on its position relative to the field.
The electric potential due to an idealized point charge is continuous in all space except at the location of the point charge itself. It is proportional to 1/r, where 'r' represents the distance from the point charge. This means that as the distance from the point charge increases, the electric potential decreases. For example, consider a hollow sphere with a charge. The potential is constant inside the sphere and follows the behaviour of an uncharged sphere outside.
Graphs of electric potential for a radial field, representing a point charge, may differ from those of gravitational potential due to differences in the reference points and the nature of the fields. In electric potential graphs, there is no corresponding radius value, suggesting that electric potential may be defined both inside and above the surface of a charge. This differs from gravitational potential, which typically starts at the radius of the mass being described.
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Electric potential and energy
Electric potential energy is a scalar quantity, meaning it has magnitude but no direction. It is measured in joules and denoted by the letter V. An object has electric potential energy due to its own electric charge and its relative position to other electrically charged objects.
The electric potential energy of an object is defined as the total work done by an external force in bringing the charge from infinity to a given point. This can be calculated using the formula:
∫ (ra→rb) F.dr = – (Ua – Ub)
Where rb is present at infinity, and ra is r. The electric potential at infinity is assumed to be zero.
The electric potential of a charge is the total work done to bring the charge from infinity to the given point, divided by the quantity of charge. In an electrical circuit, the potential between two points is defined as the work done by an external agent to move a unit charge from one point to another.
The electric potential due to an idealized point charge is continuous in all space except at the location of the point charge. The electric field is not continuous across an idealized surface charge, but it is not infinite at any point, so the electric potential is continuous across such a surface.
The SI unit of electric potential is the volt, denoted as V, in honour of Alessandro Volta. The voltmeter measures the potential difference between two different types of metal, and this quantity is called the electrochemical potential or fermi level. The pure unadjusted electric potential, V, is sometimes called the Galvani potential, ϕ.
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Frequently asked questions
Electric potential, also known as voltage, is a scalar field that gives an idea of the energy landscape produced by an electric field. It is a single number at each point in space that characterises the electric potential.
Determining the electric potential of a graph helps to understand the energy landscape surrounding the charges in question. It also helps to determine the potential difference from field-distance graphs.
The electric potential around a positive charge decreases with distance, whereas it increases with distance around a negative charge.
The equation for the electric potential of a point charge is V = 1/4πϵ0 * q/r, where V is the electric potential, q is the charge, r is the distance, and ϵ0 is the vacuum permittivity.
The graph of electric potential against distance follows a 1/r relation for a positive charge and a -1/r relation for a negative charge. The potential difference due to a charge can also be determined from the area under a field-distance graph.
