Derivation of Electric Field and Potential Relation with Diagram
Understanding the Relation Between Electric Field And Electric Potential is essential for mastering core principles in physics, particularly for students studying Class 12th physics. These concepts form the foundation for electrostatics and are crucial for analyzing how charges interact within an electric field. In this article, we will break down what electric field and electric potential mean, derive their mathematical relation, explore formulas, and clarify their interdependence using clear diagrams and explanations.
Meaning of Electric Field and Electric Potential
Electric Field (E) refers to the force per unit positive charge exerted at a point in space due to other electric charges. It describes the direction and magnitude of force that a test charge would experience. On the other hand, Electric Potential (V) at a point is the work done per unit charge to move a test charge from infinity to that point, without acceleration. Together, these quantities are central to understanding how charged particles influence each other and their surroundings.
Relation Between Electric Field And Electric Potential: Formula and Explanation
The relation between electric field and potential gradient can be derived by analyzing how the electric potential changes with respect to distance. In physics, especially in class 12, this relation helps to solve many problems involving point charges, potential differences, and field strengths.
Step-by-Step Derivation: Class 12 Physics
Let’s systematically derive the relation between electric field and electric potential:
- Electric Potential Due to a Point Charge:
The electric potential (V) at a distance r from a point charge Q is given by:
V = kQ/r, where k = 1/(4πε₀) is Coulomb’s constant. - Calculating the Potential Gradient:
The potential gradient is the rate at which potential changes with distance:
Potential Gradient = dV/dr - Differentiate the Potential:
Differentiating V = kQ/r with respect to r:
dV/dr = d/dr (kQ/r) = -kQ/r2 - Expressing Electric Field in Terms of Potential Gradient:
The electric field E due to a point charge at distance r:
E = kQ/r2 - Relating E and dV/dr:
Comparing the two expressions:
dV/dr = -E
Or, E = -dV/dr
This equation, E = -dV/dr, is the fundamental relation between electric field and electric potential. It shows that the electric field at any point equals the negative gradient (rate of change) of electric potential at that point.
Interpretation and Physical Significance
The negative sign in the relation between electric field and electric potential formula indicates that the electric field points in the direction of decreasing potential. In other words, positive charges naturally move from regions of high potential to low potential, following the electric field lines.
Diagram: Relation Between Electric Field and Electric Potential
Imagine equipotential lines (lines where V is constant) and electric field lines in space. The electric field is always perpendicular to equipotential surfaces and points toward decreasing potential. The closer these equipotential surfaces, the stronger the electric field.
- E is strongest where equipotential surfaces are closest together.
- E points from higher to lower potential.
- At every point, E = -dV/dr (in 1D; for 3D, it's the negative gradient, E = -∇V).
Table: Key Formulas Linking Electric Field and Electric Potential
| Quantity | Formula | Description |
|---|---|---|
| Electric Potential (V) | V = kQ/r | Potential at distance r from point charge Q |
| Electric Field (E) | E = kQ/r2 | Field at distance r due to Q |
| General Relation | E = -dV/dr | Electric field equals negative rate of change of potential |
These formulas are fundamental in Class 12 physics and widely used in electrostatics. To deepen your understanding of related concepts like electric field energy or the effect of electric field on conductors and insulators, you can explore more resources on electrostatics and electric fields of point charges.
Application: Electric Potential and Electric Field Energy
The relation between electric field and electric potential energy is integral to understanding how energy is stored and transferred in an electric field. When work is done to move a charge in an electric field, the change in electric potential energy equals the charge multiplied by the potential difference between initial and final points:
U = q(Vf - Vi)
And for small displacements along the electric field:
dU = qE·dr
Summary: Key Points to Remember
To quickly recap the relation between electric field and electric potential:
- Electric field intensity (E) is the negative gradient of the electric potential (dV/dr).
- E = -dV/dr, for a point charge: V = kQ/r and E = kQ/r2.
- This forms a foundation for solving electrostatic problems in class 12 physics and beyond.
For further reference on this topic and to see more derivations for class 12th physics, visit Vedantu's detailed explanation. Related subjects, such as deriving the electric field from potential and a comprehensive list of class 12 physics formulas, will support your exam preparation and deepen your understanding of physical laws.
By mastering the connection between electric field and electric potential, you'll be well-equipped to approach a range of physics problems involving energy, forces, and electric potential differences.
FAQs on Electric Field and Electric Potential: Relationship, Formula & Examples
1. What is the relation between electric field and electric potential?
The relation between electric field and electric potential is that the electric field is the negative gradient of electric potential. This means:
- Electric field (E) at a point is equal to the negative rate of change of electric potential (V) with respect to displacement.
- Mathematically, E = -dV/dx along one dimension.
- The electric field points in the direction of the greatest decrease in potential.
2. How do you mathematically express the relationship between electric field and electric potential?
The electric field is the negative gradient of the electric potential. For one dimension:
- E = -dV/dx
- E = -∇V where ∇V is the gradient of potential.
3. What is electric potential?
Electric potential is the amount of work done in bringing a unit positive charge from infinity to a point inside an electric field. Key points include:
- Measured in volts (V).
- Scalar quantity.
- Represents the 'energy per unit charge' at a specific point.
4. What is electric field?
Electric field is a region around a charged particle in which another charge experiences a force. Key features:
- Vector quantity, having both magnitude and direction.
- Measured in newtons per coulomb (N/C) or volts per metre (V/m).
- The strength and direction indicate the nature of the force a unit positive test charge would feel.
5. Why is the electric field the negative rate of change of potential?
The negative sign indicates that electric field points in the direction of decreasing potential. This is because:
- Positive charges naturally move from higher to lower potential (down the potential gradient).
- The electric field opposes the increase in potential, hence the negative rate of change.
6. How can you calculate the electric field from a given potential difference?
The electric field between two points is the potential difference divided by the distance between them. Formula:
- E = ΔV/d where ΔV is the potential difference and d is the separation.
7. What is the SI unit of electric field and electric potential?
The SI unit of electric field is newton per coulomb (N/C) or volt per metre (V/m), and the unit of electric potential is volt (V).
- Electric Field: N/C or V/m
- Electric Potential: Volt (V)
8. How does a graph of electric potential versus distance help understand the electric field?
The slope of the electric potential versus distance graph at any point gives the magnitude of the electric field at that point.
- A steep slope (rapid change in potential) indicates a strong electric field.
- If the curve is flat (no change in potential), the electric field is zero.
9. Can electric potential exist without an electric field?
Yes, electric potential can exist even if the electric field is zero.
- If the potential is uniform (constant everywhere), the electric field is zero because E = -dV/dx becomes zero.
10. What is the physical significance of the relationship between electric field and electric potential?
The relationship shows how energy changes for a charge moving in an electric field. In practical terms:
- Electric field indicates the force direction and magnitude on a charge.
- Electric potential describes the energy status of a point in the field.
- This relationship underpins many electrostatics concepts like work done, potential energy, and voltage calculations.
