Sub topics

Introduction

"Arbitrary" in physics

Electric field inside an charged body is always zero

In electrostatics, the internal electric field in a conductor is always zero — because free charges rearrange to cancel any internal field.

Conservative forces

Conservative forces are like rechargeable batteries — they store and return energy without wasting it.

Types and Examples of Conservative Forces

Non-Conservative Forces (Just for comparison):

These lose energy as heat, sound, etc.

• Friction
• Air resistance
• Applied forces (pushing/pulling) In these, you can't get back all the energy you put in.

Radial Direction

The radial direction refers to a line or path that goes straight outward or inward from a central point — like a radius of a circle or sphere.

Tangential Direction (The Opposite)

• Tangential direction is perpendicular to the radial direction.
• It goes around the center — like moving along the edge of a circle instead of toward or away from it.
• the opposite of radial is typically called the tangential or transverse direction, which means moving sideways to the radial path — around, not in or out.

Work and Potential Energy

• Whenever you move a charge against the field, you're increasing its electric potential energy, so you do work.
• When you move along the field, potential energy decreases, and the field does work for you.

Formulas for work done

Law of cosines

The Law of Cosines (also known as the Cosine Rule) is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is essentially a more general version of the Pythagorean theorem that works for all triangles, not just right-angled ones.

The Formula

For a triangle with sides $a, b,$ and $c$, and the angle opposite to side $c$ being $\gamma$ (or $C$):

$c^2 = a^2 + b^2 - 2ab \cos(\gamma)$

Similarly, it can be written for the other sides:

• $a^2 = b^2 + c^2 - 2bc \cos(\alpha)$

• $b^2 = a^2 + c^2 - 2ac \cos(\beta)$

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Why is it useful?

You can use the Law of Cosines in two primary scenarios:

1. SAS (Side-Angle-Side): When you know two sides and the angle between them, you can find the third side.

2. SSS (Side-Side-Side): When you know all three sides, you can find any of the angles.

Relation to the Pythagorean Theorem

If the angle $\gamma$ is exactly $90^\circ$ (a right triangle), the formula becomes:

$c^2 = a^2 + b^2 - 2ab \cos(90^\circ)$

Since $\cos(90^\circ) = 0$, the last term disappears, leaving you with the familiar Pythagorean theorem:

$c^2 = a^2 + b^2$

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Derivation (Using Coordinates)

One of the simplest ways to derive the law is by placing a triangle on a coordinate plane:

1. Place vertex $C$ at the origin $(0,0)$.

2. Place side $b$ along the x-axis, so vertex $A$ is at $(b, 0)$.

3. Using trigonometry, vertex $B$ is at $(a \cos \gamma, a \sin \gamma)$.

4. Use the distance formula to find the length of side $c$ (the distance between $A$ and $B$):

   $c^2 = (a \cos \gamma - b)^2 + (a \sin \gamma - 0)^2$

5. Expand the squares:

   $c^2 = a^2 \cos^2 \gamma - 2ab \cos \gamma + b^2 + a^2 \sin^2 \gamma$

6. Group the terms:

   $c^2 = a^2(\cos^2 \gamma + \sin^2 \gamma) + b^2 - 2ab \cos \gamma$

7. Since $\cos^2 \gamma + \sin^2 \gamma = 1$:

   $c^2 = a^2 + b^2 - 2ab \cos \gamma$

Application in Physics

As seen in your uploaded documents (specifically the derivation for Electric Dipole Potential), the Law of Cosines is used to determine the distance from a point to each charge in a dipole.

If a point $P$ is at a distance $r$ from the center of a dipole of length $2a$, the distances $r_1$ and $r_2$ to the charges are found using:

• $r_1^2 = r^2 + a^2 - 2ra \cos \theta$

• $r_2^2 = r^2 + a^2 + 2ra \cos \theta$ (using the supplementary angle)

1. Electrostatic potential

2. Potential due to a point charge

3. Potential due to an electric dipole

4. Potential due to a system of charges

5. Equipotential surfaces

6. Potential energy of a system of charges

7. Potential energy in an external field

8. Electrostatics of conductors

Applications of Electrostatic shielding

9. Dielectrics and polarisation

Permanent Dipole Moment

The permanent dipole moment is an intrinsic property of individual polar molecules.

It arises from the asymmetric distribution of positive and negative charges within a molecule due to differences in electronegativity and molecular geometry.

This dipole moment exists without the need for an external electric field.

Example: In a water (H₂O) molecule, electrons spend more time near the oxygen atom than the hydrogen atoms, creating a permanent charge separation and thus a permanent dipole moment

Net Dipole Moment (in a dielectric material)

The net dipole moment is the vector sum of all individual molecular dipole moments within a bulk material (like a dielectric).

In the absence of an external electric field, the permanent dipoles in a polar dielectric are randomly oriented due to thermal motion, so their dipole moments cancel out, resulting in a net dipole moment of zero at the macroscopic scale.

When an external electric field is applied, these permanent dipoles tend to align partially along the field direction, causing a nonzero net dipole moment in the material. The extent of alignment depends on the field strength and temperature.

This induced alignment of permanent dipoles is called orientation polarization, and it leads to the polarization of the dielectric as a whole.

Difference between net dipole moment and permanent dipole moment

Permanent dipole moment in a molecule

H (+) O (-) H (+) \ | / — Dipole moment vector μ (from – to +)

Random orientation without field

Molecules' dipoles point randomly:

← ↑ → ↓ ← → → ↓ ↑ ← ↑ ↓ Result: Net dipole moment ≈ 0

Partial alignment in the presence of external field

Applied field → → →

Dipoles partially align: → → ↑ → → ↓ →

Net dipole moment points in direction of applied field

10. Capacitors and Capacitance

In Cameras

Capacitors are essential to the function of camera flash systems and power supply management:

1. Flash Charging and Discharge

Energy Storage for Flashes: Capacitors store electrical energy over a short period (charged slowly from the battery) and then release it almost instantaneously to power the camera flash. This sudden release produces the high-intensity burst of light needed for taking photographs in low light.

The rapid energy release is possible because capacitors can discharge much faster than batteries, delivering a quick pulse without draining the battery immediately.

2. Power Supply Stabilization

Capacitors help smooth out voltage supplied to sensitive electronic components (sensors, processors) in cameras. This stabilization prevents malfunctions or distortions in digital images that might result from voltage fluctuations.

11. The parallel plate capacitor

12. Effect of Dielectric on capacitance

13. Combination of capacitors

14. Energy stored in a capacitor