How to Derive the Formula Relating Electric Field and Electric Potential?
The relationship between electric field and electric potential is fundamental in electrostatics, helping us understand how charges affect their surroundings and how energy changes occur during interactions. Understanding these concepts is essential for grasping the behavior of electric charges, predicting motion, and solving Physics problems related to electrostatics.
Definition of Electric Field and Electric Potential
An electric field is the region around a charged particle or object in which another charge experiences an electric force. Every charge produces its own electric field. The intensity of the electric field at a point is the force acting on a unit positive test charge placed at that point. It is a vector quantity, meaning it has both magnitude and direction.
An electric potential at a point is the work done by an external agent in moving a unit positive charge from infinity (reference point) to that point, without changing its kinetic energy. It helps measure the potential energy per unit charge at a specific location in the field. Unlike the electric field, electric potential is a scalar quantity.
Key Characteristics and Units
| Quantity | Definition | SI Unit | Nature |
|---|---|---|---|
| Electric Field (E) | Force per unit charge | N/C or V/m | Vector |
| Electric Potential (V) | Work per unit charge | Volt (V) | Scalar |
Relationship Between Electric Field and Electric Potential
The electric field at a point is related to the electric potential by the negative gradient:
This means the electric field points in the direction where the potential decreases most rapidly. The negative sign shows that a test charge moves spontaneously from higher to lower potential under the influence of the field.
Electric Potential Due to a Point Charge
The potential at a distance r from a point charge q is:
The electric field at the same location is:
As can be seen, the electric field is the spatial rate of change of potential.
Step-by-Step Approach to Problem Solving
| Step | Description | Key Tip |
|---|---|---|
| 1 | Identify what is given (potential, field, position, charge). | Write known values and required unknowns. |
| 2 | Choose the suitable formula based on what is required. | Use E = -dV/dx or V = (1/4πε0)(q/r) appropriately. |
| 3 | Substitute values and calculate carefully. | Be careful with units and signs. |
| 4 | Interpret the result and state the answer clearly. | Box the final answer with units. |
Application Example
Example: Given the electric potential V(x) = 5 - 3x (in volts), find the electric field at x = 2 m.
Solution:
Find the derivative: dV/dx = -3.
So, E = -dV/dx = -(-3) = 3 V/m.
The electric field at x = 2 m is 3 V/m.
Difference Between Electric Field and Electric Potential
| Aspect | Electric Field (E) | Electric Potential (V) |
|---|---|---|
| Definition | Force per unit charge at a point | Potential energy per unit charge |
| Quantity Type | Vector | Scalar |
| SI Unit | N/C or V/m | V (Volt) |
| Direction | Has both magnitude and direction | Only magnitude |
| Role | Describes force on a charge | Describes energy per charge |
Sample Problem
Q: If electric potential V = 2x2 + 3y at a point, find the electric field at (1, 1, 0).
Solution:
E = -dV/dx i - dV/dy j
dV/dx = 4x; dV/dy = 3
At (1, 1, 0): E = -4i - 3j
Key Concepts to Remember
- The electric field always points from higher to lower electric potential.
- The field is perpendicular to equipotential surfaces.
- If the potential function is known, the field can be found by differentiation; if the field is known, the potential can be found by integration.
Further Learning on Vedantu
- Electric Potential Due to Point Charge
- Electric Field Due to Point Charge
- Electrostatics Concepts
- Electric Potential of a Dipole
In Summary
- Electric field and electric potential are closely linked but distinct concepts.
- Mastering their relationship is essential for scoring well in Physics and understanding electrostatics problems.
- Consistent practice and conceptual clarity will help you solve a variety of numerical and theoretical questions.
FAQs on Understanding the Relation Between Electric Field and Electric Potential
1. What is the relation between electric field and electric potential?
The electric field (E) is the negative gradient of electric potential (V). This means the electric field at a point shows the rate at which potential decreases with distance.
Mathematically: E = -dV/dx
2. How do you derive the formula E = -dV/dx?
To derive E = -dV/dx:
- The work done (W) in moving a unit positive charge through a potential difference (dV) is dV = -E dx
- Rearrange to get the electric field: E = -dV/dx, where the negative sign indicates the direction of decreasing potential.
3. What does the negative sign in E = -dV/dx signify?
The negative sign shows that the electric field points in the direction of decreasing electric potential. In other words, electric field lines always go from higher to lower potential regions.
4. What is electric potential and how is it different from electric field?
Electric potential (V) is a scalar quantity representing the work done per unit charge to bring a test charge from infinity to a point.
Electric field (E) is a vector quantity representing the force experienced by a unit positive charge at a point.
- Electric field: has both magnitude and direction (vector)
- Electric potential: has only magnitude (scalar)
5. Can you explain the difference between electric potential and electric potential energy?
Electric potential (V) refers to the work done per unit charge to move a test charge from infinity to a point.
Electric potential energy (U) is the total energy a charge possesses due to its position in an electric field: U = qV, where q is the charge.
- Potential is per unit charge; potential energy is for the whole charge
6. How are potential and electric field related for a point charge?
For a point charge (q) at distance r:
- Electric potential: V = (1/4πε₀)(q/r)
- Electric field: E = (1/4πε₀)(q/r²)
- Using E = -dV/dr, you can see that the electric field is the rate of decrease of potential with respect to distance from the charge.
7. What is a potential gradient?
The potential gradient is the rate of change of electric potential with respect to position: dV/dx. The electric field at a point is numerically equal to the potential gradient, but acts in the opposite direction: E = -dV/dx.
8. Does electric potential always decrease in the direction of the electric field?
Yes, in electrostatics, electric potential always decreases in the direction of the electric field. This is why the electric field lines point from high potential to low potential regions.
9. How can you calculate the electric potential difference between two points if you know the electric field?
If the electric field (E) is known, the potential difference (ΔV) between two points A and B can be calculated by integrating:
ΔV = V(B) - V(A) = -∫AB E·dx
This formula shows potential decreases along the direction of E.
10. What are equipotential surfaces and how do they relate to electric fields?
Equipotential surfaces are surfaces on which the electric potential is the same everywhere.
- No work is required to move a charge on an equipotential surface
- Electric field lines are always perpendicular to equipotential surfaces
11. Why is understanding the relation between electric field and potential important for Physics exams?
Mastering the relation between electric field and potential helps in:
- Solving numerical problems quickly and accurately
- Tackling conceptual questions in JEE, NEET, and CBSE exams
- Understanding the core principles of electrostatics for further physics applications
12. How do you identify whether to use E = -dV/dx or E = (1/4πε₀)(q/r²) in problems?
Use E = -dV/dx when the electric potential is given as a function of position and you need the field at a point. Use E = (1/4πε₀)(q/r²) when dealing with the field due to a point charge at a distance r. Choose the formula based on the information provided in the question.
