Potential due to Point Charge — Revision Notes

⚡ 30-Second Revision

Definition: — Work done by external agent to bring unit positive charge from infinity to a point.
Formula: — $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} = \frac{kQ}{r}$
Units: — Volt (V) or J/C
Nature: — Scalar quantity (includes sign of Q)
Sign: — Positive for +Q, Negative for -Q
Dependence: — $V \propto 1/r$
Superposition: — $V_{total} = \sum V_i = \sum \frac{kQ_i}{r_i}$ (algebraic sum)
Work Done: — $W_{ext} = q(V_{final} - V_{initial})$
Relationship with E: — $E = -dV/dr$ (in 1D), $r = V/E$ (for point charge)
Equipotential Surfaces: — Concentric spheres for point charge; $W=0$ along them.

2-Minute Revision

Electric potential due to a point charge is a scalar measure of the electric environment, defined as the work required per unit positive charge to bring it from infinity to a specific point. The fundamental formula is $V = kQ/r$, where $Q$ is the source charge (with its sign) and $r$ is the distance.

Remember that $k = 9 \times 10^9,\text{N m}^2/\text{C}^2$. A positive charge creates positive potential, and a negative charge creates negative potential. This scalar nature is a huge advantage when dealing with multiple charges, as the total potential is simply the algebraic sum of individual potentials ($V_{total} = sum V_i$).

This is the superposition principle. Crucially, potential varies as $1/r$, unlike the electric field which varies as $1/r^2$. The work done by an external agent to move a charge $q$ from point A to B is $W = q(V_B - V_A)$.

Equipotential surfaces for a point charge are concentric spheres, and no work is done moving a charge along such a surface. Always pay attention to signs and units in calculations.

5-Minute Revision

Let's consolidate the concept of electric potential due to a point charge, a cornerstone of electrostatics. Electric potential ($V$) at a point is fundamentally the work done by an external force to bring a unit positive test charge from infinity to that point without acceleration. It's a scalar quantity, meaning it has magnitude and sign, but no direction, simplifying calculations significantly. The standard unit is the Volt (V), equivalent to Joules per Coulomb (J/C).

For a single isolated point charge $Q$, the potential at a distance $r$ is given by $V = \frac{kQ}{r}$, where $k = \frac{1}{4\pi\epsilon_0} \approx 9 \times 10^9,\text{N m}^2/\text{C}^2$. The sign of $Q$ is critical: a positive charge creates a positive potential, and a negative charge creates a negative potential.

This implies that for a positive charge, you do positive work to bring a positive test charge closer, while for a negative charge, the field itself does positive work (you do negative work) to bring a positive test charge closer.

The superposition principle is vital: for a system of multiple point charges, the total potential at any point is the algebraic sum of the potentials due to each individual charge. $V_{total} = V_1 + V_2 + dots + V_n = \sum \frac{kQ_i}{r_i}$. This is much simpler than vector addition for electric fields.

Work done is directly linked to potential difference: the work done by an external agent to move a charge $q$ from point A to point B is $W_{AB} = q(V_B - V_A)$. If $W_{AB}$ is positive, the external agent does work; if negative, the electric field does work. Remember that $W_{field} = -W_{ext}$.

Key distinctions to remember for NEET: Potential varies as $1/r$, while the electric field varies as $1/r^2$. Potential is scalar, field is vector. For a point charge, equipotential surfaces are concentric spheres, and electric field lines are always perpendicular to these surfaces. No work is done in moving a charge along an equipotential surface.

Example: Two charges, $Q_1 = +4,\text{nC}$ at $(0,0)$ and $Q_2 = -6,\text{nC}$ at $(0,3,\text{m})$. Find potential at $(4,\text{m},0)$.

1
Distance $r_1$ from $Q_1$ to $(4,0)$ is $sqrt{4^2+0^2} = 4,\text{m}$.
2
Distance $r_2$ from $Q_2$ to $(4,0)$ is $sqrt{4^2+3^2} = sqrt{16+9} = sqrt{25} = 5,\text{m}$.
3
$V_1 = \frac{(9 \times 10^9)(+4 \times 10^{-9})}{4} = +9,\text{V}$.
4
$V_2 = \frac{(9 \times 10^9)(-6 \times 10^{-9})}{5} = -10.8,\text{V}$.
5
$V_{total} = V_1 + V_2 = 9,\text{V} - 10.8,\text{V} = -1.8,\text{V}$.

Prelims Revision Notes

Potential due to Point Charge (PHY-11-06) - NEET Revision Notes

1. Definition and Basic Formula:

Electric Potential (V): — Work done by an external agent to bring a unit positive test charge ($q_0$) from infinity ($infty$) to a specific point P in an electric field, without acceleration. $V = W_{infty \to P} / q_0$.
Formula for Point Charge Q: — $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} = \frac{kQ}{r}$

* $k = 9 \times 10^9,\text{N m}^2/\text{C}^2$ (Coulomb's constant) * $Q$: Source charge (include its sign!) * $r$: Distance from the point charge to the point of interest.

2. Key Properties:

Scalar Quantity: — Potential has magnitude and sign, but no direction. This simplifies calculations for multiple charges.
Units: — Volt (V) or Joules per Coulomb (J/C).
Sign Convention:

* Positive potential ($V > 0$) for a positive source charge ($Q > 0$). * Negative potential ($V < 0$) for a negative source charge ($Q < 0$).

Reference Point: — Potential at infinity is conventionally taken as zero ($V_{\infty} = 0$).
Dependence on Distance: — $V \propto 1/r$. It decreases less rapidly with distance than the electric field ($E \propto 1/r^2$).

3. Superposition Principle:

For a system of $n$ point charges ($Q_1, Q_2, \dots, Q_n$) at distances ($r_1, r_2, \dots, r_n$) from a point P, the total potential at P is the algebraic sum of individual potentials:

$V_{total} = V_1 + V_2 + \dots + V_n = \sum_{i=1}^{n} \frac{kQ_i}{r_i}$ * Crucial: Always include the sign of each charge $Q_i$.

4. Work Done and Potential Difference:

Potential Difference ($Delta V$): — $V_B - V_A = W_{ext} / q_0$.
Work Done by External Agent ($W_{ext}$): — To move charge $q$ from A to B: $W_{AB} = q(V_B - V_A)$.
Work Done by Electric Field ($W_{field}$): — $W_{field} = -W_{ext} = q(V_A - V_B)$.

* If $W_{ext} > 0$, external agent does work (e.g., pushing +ve charge towards +ve source). * If $W_{ext} < 0$, electric field does work (e.g., +ve charge moving towards -ve source).

5. Equipotential Surfaces:

Surfaces where the electric potential is constant.
For a point charge, equipotential surfaces are concentric spheres centered on the charge.
Property: — No work is done in moving a test charge along an equipotential surface ($W = q\Delta V = q \times 0 = 0$).
Electric field lines are always perpendicular to equipotential surfaces.

6. Relationship with Electric Field (E):

In 1D: $E = -dV/dr$. The electric field points in the direction of decreasing potential.
For a point charge, $E = k|Q|/r^2$ and $V = kQ/r$. Thus, $E = |V|/r$ (magnitude relationship).

7. Common Mistakes to Avoid:

Forgetting the sign of the charge $Q$ in potential calculations.
Confusing potential ($1/r$) with electric field ($1/r^2$) dependence on distance.
Confusing scalar addition for potential with vector addition for electric field.
Incorrectly calculating potential difference or work done (sign errors).
Not converting units (nC to C, cm to m).

Vyyuha Quick Recall

To remember the potential formula and its properties:

Very Positive Charges Radiate Potential, Negative Charges Attract Negative Potential.

Very: Voltage (Potential)
Positive Charges: $Q$ is positive
Radiate Potential: $V$ is positive
Negative Charges: $Q$ is negative
Attract Negative Potential: $V$ is negative

And for the formula: Very Kool Quick Review: $V = kQ/r$