How Does the Electric Field Due to a Point Charge Depend on Distance?
Understanding the Electric Field Due To Point Charge is crucial in Physics, as it reveals how charges influence their surroundings and interact with other charges. Discover formulas, detailed derivations, vector concepts, and practical examples—everything you need for class 12 CBSE, competitive exams, or curiosity. Explore how the electric field changes with distance and more as you read on.
What Is the Electric Field Due To Point Charge?
An electric field is a region around a charged particle where another electric charge experiences a force. When a point charge, such as an electron or proton, is placed in space, it generates its own electric field. Any nearby charge within this region will feel an electrostatic force—either attraction or repulsion—depending on the nature (positive or negative) of the charges. For example, rubbing a glass rod with silk leaves the rod charged, and it can attract paper bits because of its electric field.
This invisible “influence” is the cornerstone of how charges interact—even when not touching. The concept of the electric field, introduced by Michael Faraday, forms the basis for much of classical electromagnetism, underpinning topics from electric circuits to forces between particles.
Electric Field: Definition and Key Features
The electric field due to a point charge is defined as the force per unit positive test charge placed at a specific point in space. It represents how a unit charge would be affected if placed within the field. The electric field’s direction is always along the force experienced by a positive test charge—outward from a positive source, and inward for a negative source. Notably, the field's magnitude depends on the distance $r$ from the charge, charge value $Q$, and the medium’s permittivity.
- It is a vector quantity (has both magnitude and direction).
- Unit: newton per coulomb (N/C) or volt per meter (V/m). See related info in Electric Field Unit.
- The field becomes weaker as the distance from the point charge increases.
- Not uniform: The electric field due to point charge is non-uniform, since its value changes with $r$.
Electric Field Due To Point Charge Formula
The electric field due to a point charge formula is given by:
Where:
- $E$ = electric field intensity (N/C)
- $Q$ = point charge (Coulombs)
- $r$ = distance from point charge (meters)
- $\epsilon_0$ = permittivity of free space ($8.854 \times 10^{-12}~{\rm C^2/N{\cdot}m^2}$)
This relationship shows the electric field due to point charge depends on distance $r$ as the inverse square: $E \propto \frac{1}{r^2}$. Doubling the distance reduces field strength to one-fourth. In vector form, the formula becomes:
where $\hat{r}$ is the unit vector from the charge to the observation point. Explore practical formulas for class 12 at Physics Formulas for Class 12.
Step-by-Step Derivation: Electric Field Due To Point Charge (Class 12)
- Consider a point charge $Q$ at the origin $O$.
- Select point $P$ at distance $r$ from $O$.
- Place a test charge $q_0$ at $P$.
- By Coulomb's law, the force on $q_0$ is $F = \frac{1}{4\pi\epsilon_0} \frac{|Q||q_0|}{r^2}.$
- Electric field intensity at $P$ is the force per unit positive charge: $E = \frac{F}{q_0}$.
- Substitute $F$ from above: $E = \frac{1}{4\pi\epsilon_0} \frac{|Q|}{r^2}$.
This straightforward derivation is essential for board exams and is often referenced in important class 12 Physics derivations.
Electric Field Dependence on Distance r
The electric field due to point charge graph shows a rapid fall in field strength as $r$ increases. Mathematically, $E \propto \frac{1}{r^2}$. This “inverse square law” means small changes in distance make a big difference. This non-uniform nature is key: close to the charge, the field is very strong; farther away, it becomes weaker. Learn more about inverse-square laws at Inverse Square Law.
Electric Field Due To Multiple Point Charges (Superposition Principle)
When several charges are present, their fields combine. The net electric field at any point is the vector sum of fields by each charge. For charges $Q_1, Q_2, ..., Q_n$ at positions $\vec{r}_1, \vec{r}_2, ..., \vec{r}_n$:
Here, $\hat{r}_i$ points from $Q_i$ to the observation point $\vec{r}$. This method is widely used to calculate the field in systems of discrete charges.
Applications and Numerical Example
Let’s use the electric field formula in a real scenario:
- Calculating the intensity near a charged object (like a Van de Graaff generator)
- Understanding repulsion/attraction of small particles (as in electroscopes—see Electroscope)
- Modeling fields in electrostatic experiments and sensors
Example Calculation: A charge $Q = 2 \times 10^{-6}$ C is situated in air. What is the electric field at a point $r = 0.3$ m away?
- Use $$ E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} $$ with $k = 9 \times 10^9~{\rm Nm^2C^{-2}}$.
- Plug in values: $E = 9 \times 10^9 \cdot \frac{2 \times 10^{-6}}{0.3^2}$.
- Calculate: $E = 9 \times 10^9 \cdot \frac{2 \times 10^{-6}}{0.09} = 9 \times 10^9 \times (2.22 \times 10^{-5}) = 2 \times 10^5~{\rm N/C}$.
Thus, the field strength at $0.3\,m$ is $2 \times 10^5\,{\rm N/C}$, pointing away if the charge is positive.
Vector Form & Field Diagrams
Electric field due to point charge in vector form is crucial for 3D calculations. Its direction and magnitude are fully described using a unit vector $\hat{r}$. Diagrams often show field lines radiating outward from a positive point charge and inward for a negative one—these help visualize how test charges would move. The field is strongest near the charge where lines are densest. For a detailed visual representation, see Electric Field Due To Point Charge Diagram.
Using Gauss’s Law: Alternative Approach
Electric field due to point charge using Gauss law is another way to derive the same formula, especially in cases of high symmetry. Gauss’s law states that the total electric flux through a closed surface surrounding a charge $Q$ equals $\frac{Q}{\epsilon_0}$. Applying Gauss's law to a spherical surface centered on the charge, you again arrive at:
This confirms the same $1/r^2$ dependence as found through Coulomb’s law. Read about related applications and comparisons at Superposition Principle.
Electric Field Due To Point Charge in Different Languages & Contexts
For learners in regional languages, the concept is known as इलेक्ट्रिक फील्ड ड्यू टू पॉइंट चार्ज in Hindi and is commonly featured in class 12 Physics resources. The formula and explanations remain universally applicable, regardless of language.
Summary Table: Electric Field Due To Point Charge – At a Glance
| Parameter | Symbol/Value | Unit |
|---|---|---|
| Point Charge | $Q$ | Coulombs (C) |
| Distance from Charge | $r$ | Meter (m) |
| Permittivity (vacuum) | $\epsilon_0$ | $8.854 \times 10^{-12}~{\rm C^2/N{\cdot}m^2}$ |
| Electric Field Intensity | $E$ | Newton/Coulomb (N/C) |
| Formula | $E = \dfrac{1}{4\pi\epsilon_0} \dfrac{Q}{r^2}$ | - |
This table summarizes the core variables and relationships for the electric field due to point charge, which is central to many physics problems and competitive exams like NEET and JEE.
In conclusion, mastering the Electric Field Due To Point Charge, including its formula, vector nature, distance dependence, and derivations, will deepen your understanding of action-at-a-distance phenomena. Practice with problems and explore topics like the field due to a charged wire or basic electrostatics for broader insights and exam success.
FAQs on Electric Field Due to a Point Charge – Formula, Derivation, and Explanation
1. What is the electric field due to a point charge?
The electric field due to a point charge is the region around the charge where another charge experiences a force. It is mathematically given by Coulomb's law:
- The formula is E = (1/4πε₀)·(q/r²), where E is the electric field, q is the charge, and r is the distance from the point charge.
- The direction of the field is away from the charge if it is positive, and towards the charge if it is negative.
- This concept is fundamental in understanding electric force and field lines in physics.
2. State the formula for the electric field produced by a point charge.
The formula for the electric field produced by a point charge is:
- E = (1/4πε₀) × (q/r²)
- Where E is the electric field intensity, q is the magnitude of the point charge, r is the distance from the charge, and ε₀ is the permittivity of free space.
3. What is the direction of the electric field due to a positive point charge?
The electric field due to a positive point charge points radially outward from the charge:
- Field lines move away from the positive charge in all directions.
- This direction is determined using a small positive test charge placed in the field.
4. How does the electric field change with distance from a point charge?
The electric field due to a point charge decreases inversely with the square of the distance.
- If the distance (r) is doubled, the electric field (E) becomes one-fourth.
- This is known as the inverse square law, expressed in the formula E ∝ 1/r².
5. Explain the characteristics of electric field lines due to a point charge.
Electric field lines due to a point charge have distinct and important characteristics:
- They radiate symmetrically from the charge.
- Lines diverge from a positive charge and converge on a negative charge.
- No two field lines ever intersect.
- The density of lines indicates the strength of the field (closer = stronger).
6. What is meant by the term 'unit positive test charge' in electric field concepts?
A unit positive test charge is a hypothetical charge of +1 coulomb used to define the direction and magnitude of the electric field:
- It helps measure the force and direction at any point in an electric field.
- This ensures the test charge does not disturb the existing field.
7. Write two differences between electric field and electric field intensity.
Electric field and electric field intensity are related but distinct concepts:
- Electric field refers to the region where electric force acts on a charge.
- Electric field intensity (E) is the quantitative measure, defined as force per unit charge.
- Electric field is a qualitative concept (region), while intensity is a vector value with magnitude and direction.
8. How do you calculate the force experienced by a charge placed in the electric field of a point charge?
The force experienced by a charge in the field of a point charge is given by:
- F = q'E, where F is the force, q' is the test charge, and E is the electric field at that point.
- This formula relates the magnitude of the field and the amount of test charge present.
9. What are the SI units of electric field?
The SI unit of electric field is newton per coulomb (N/C) or volt per meter (V/m):
- Electric field (E) = Force (N) / Charge (C)
- This unit relates the strength of the field to force exerted on a unit charge.
10. What happens to the electric field if the magnitude of the point charge increases?
If the magnitude of the point charge increases, the electric field at a given distance also increases proportionally:
- The electric field E is directly proportional to the charge q: E ∝ q
- Doubling the charge doubles the field strength at all points around the charge.
11. Define electric field intensity at a point due to a point charge.
Electric field intensity at a point due to a point charge is the force experienced by a unit positive charge placed at that point due to the presence of the point charge:
- It is given by E = (1/4πε₀)·(q/r²), where q is the charge and r is the distance from the point charge.
12. Why do electric field lines never intersect?
Electric field lines never intersect because at any given point, the electric field has a unique direction:
- If lines intersected, a test charge would experience two different forces at the same point, which is impossible.
- This ensures the field is well-defined everywhere around a charge.
