Uniform E-field & V

\[\Delta V = -\vec E \cdot \Delta \vec r\]

Variable Definitions

\( \Delta V \) change in electric potential
\( \vec E \) uniform electric field
\( \Delta \vec r \) displacement between points

Applications

CBSE SQP Q1

Uniform field \( \vec E = E\hat i \). Compare potentials at A(0,0), B(1 cm,0), C(0,1 cm).

Key Idea

Potential drops along field lines, unchanged along equipotentials.

Answer

\(V_A > V_B\) and \(V_C = V_A\).

Quick Check

👍/👎 Does \(V_C = V_A\)?

Source: CBSE Sample Paper 2024-25

Connected Spheres

CBSE SQP Q2: Equal potential ⇒ compare surface fields.

1

Charge Redistribution

Wire joins two distant metal spheres; charges flow until both sit at the same potential \(V\).

2

Equal-Potential Condition

\(kQ_1/r_1 = kQ_2/r_2\)  ⇒  \(Q_1/Q_2 = r_1/r_2\).

3

Surface Field Ratio

With \(E = kQ/r^{2}\), we get \(E_1/E_2 = r_2/r_1\). The smaller sphere feels the stronger field (Option B).

Pro Tip:

Equal potential does not imply equal field—the field scales inversely with radius here.

Field of a Thick Wire

\[ B(r)= \begin{cases} \dfrac{\mu_{0} I r}{2\pi a^{2}}, & r\le a \\[6pt] \dfrac{\mu_{0} I}{2\pi r}, & r\ge a \end{cases} \]

Variable Definitions

\(I\) Steady current in wire
\(a\) Wire radius
\(r\) Distance from centre
\( \mu_{0} \) Permeability of free space

Applications

Inside vs Outside

Inside, \(B\propto r\); cross the surface and \(B\propto 1/r\). Ampere’s law applies in both zones.

CBSE Q3 Prompt

Find \( |B| \) at \(1.5a\) ( \(a/2\) above surface ) and at \(0.5a\) (inside). Give the ratio.

Answer

\(B_{\text{out}} = \mu_{0}I/3\pi a\), \(B_{\text{in}} = \mu_{0}I/4\pi a\) ⇒ ratio \(4:3\).

Quick Quiz

Vote: 1 : 1, 4 : 3, or 3 : 4 — which is correct?

Source: CBSE Sample Question Paper 2024-25 (Q3)

Diffraction Happens

cd_diffraction_rainbow

Rainbow diffraction on a CD

All waves can bend around edges

Your voice curving round a corner and the rainbow sheen on a CD are both produced by diffraction.

The effect appears whenever the size of an opening or obstacle matches the wavelength.

Key Points:

  • Diffraction is a universal property of waves—sound, light, ultrasonic, all.
  • CBSE SQP 2024-25 Q4 answer: Option D – sound waves as well as light waves.

Things to Remember

Recall these must-know formulas & patterns for Q1–Q4

Q1 Uniform E Field

Potential drops linearly: \( \Delta V = -E\,\Delta x \).

Q2 Connected Conductors

At equilibrium \( V_1 = V_2 \); surface field varies as \( E \propto 1/r \).

Q3 Thick Wire Current

Magnetic field: inside \( B \propto r \); outside \( B \propto 1/r \).

Q4 Diffraction Rule

Noticeable when aperture ≈ wavelength—true for all waves.