RC Circuits Steady State:

Steady state after long time ($t \gg \tau$):

• Capacitor fully charged
• Current through capacitor: $I_C = 0$
• Capacitor acts as open circuit
• Voltage across capacitor: $V_C = V_{applied}$

DC circuit behavior:

• Initially: capacitor uncharged, acts like short circuit
• Final state: capacitor charged, acts like open circuit
• Time constant $\tau = RC$ determines transition time

Charging and Discharging Behavior:

Charging (RC circuit with battery):

Charge as function of time: $q(t) = Q_{max}(1 - e^{-t/\tau})$

Voltage across capacitor: $V_C(t) = V_0(1 - e^{-t/\tau})$

Current: $I(t) = \frac{V_0}{R}e^{-t/\tau}$

Discharging (RC circuit without battery):

Charge: $q(t) = Q_0 e^{-t/\tau}$

Voltage: $V_C(t) = V_0 e^{-t/\tau}$

Current: $I(t) = -\frac{V_0}{R}e^{-t/\tau}$

Time constant: $\tau = RC$

At $t = \tau$: reaches ~63% of final value At $t = 5\tau$: essentially at steady state (~99%)

Angle of incidence equals angle of reflection:

$\theta_i = \theta_r$

All angles measured from normal (perpendicular to surface)

Properties:

• Incident ray, reflected ray, and normal all in same plane
• Reversible path

Virtual image formed behind mirror.

Properties:

• Image distance = object distance ($d_i = d_o$)
• Image is virtual (light doesn't actually converge there)
• Image is upright (not inverted)
• Image is same size as object (magnification = 1)
• Left-right reversal (but not up-down)

Image location: $d_i = d_o$

Spherical Mirrors Concave/Convex:

Concave mirrors (converging):

• Reflecting surface curves inward (like cave)
• Light rays converge after reflection
• Focal point is real (light actually meets)
• f > 0, R > 0

Convex mirrors (diverging):

• Reflecting surface curves outward
• Light rays diverge after reflection
• Focal point is virtual (behind mirror)
• f < 0, R < 0

Both follow same equation but with different sign conventions.

Concave mirrors (converging):

• Reflecting surface curves inward (like cave)
• Light rays converge after reflection
• Focal point is real (light actually meets)
• f > 0, R > 0

Convex mirrors (diverging):

• Reflecting surface curves outward
• Light rays diverge after reflection
• Focal point is virtual (behind mirror)
• f < 0, R < 0

Both follow same equation but with different sign conventions.

For spherical mirrors:

$f = \frac{R}{2}$

Where:

• f = focal length
• R = radius of curvature

Relationship derived from geometry for small angles (paraxial approximation).

Focal point: point where parallel rays converge (or appear to diverge from).

Relationship between angles when light passes between media:

$n_1\sin\theta_1 = n_2\sin\theta_2$

Where:

• $n_1, n_2$ = refractive indices
• $\theta_1$ = angle of incidence
• $\theta_2$ = angle of refraction

Angles measured from normal.

Ratio of speed of light in vacuum to speed in medium:

$n = \frac{c}{v}$

Where:

• c = 3.00 × 108 m/s (speed in vacuum)
• v = speed in medium

Values:

• Vacuum: n = 1
• Air: n ≈ 1.0003
• Water: n = 1.33
• Glass: n ≈ 1.5
• Diamond: n = 2.42

Behavior:

• n > 1: light slows down
• Entering denser medium (higher n): bends toward normal
• Entering less dense medium: bends away from normal

Occurs when light tries to go from dense to less dense medium at angle greater than critical angle.

Conditions:

• Light going from higher n to lower n
• Angle of incidence > critical angle

Critical angle: $\sin\theta_c = \frac{n_2}{n_1}$ (where $n_1 > n_2$$)

When $\theta_i > \theta_c$: total internal reflection, no transmission.

Applications:

• Fiber optics
• Prisms in binoculars
• Sparkling of diamonds

Angle of incidence at which refraction angle = 90 degrees :

$\sin\theta_c = \frac{n_2}{n_1}$

Only defined when $n_1 > n_2$ (dense to less dense).

Values:

• Water to air: $\theta_c = 48.8 degrees$
• Glass to air: $\theta_c = 41.8 degrees$

Converging (convex) lens:

• Thicker in middle than edges
• Brings parallel rays to focal point
• f > 0
• Forms real images

Diverging (concave) lens:

• Thinner in middle than edges
• Spreads parallel rays as if from focal point
• f < 0
• Forms virtual images

Both types follow thin lens equation with appropriate sign conventions.

Graphical method for locating images using principal rays.

Rules for lenses:

1. Ray parallel to axis -> refracts through focal point
2. Ray through focal point -> refracts parallel to axis
3. Ray through center -> continues straight (no deviation)

Rules for mirrors:

1. Ray parallel to axis -> reflects through focal point
2. Ray through focal point -> reflects parallel to axis
3. Ray through center of curvature -> reflects back on itself

Draw at least two rays; intersection gives image location.

Real images:

• Light rays actually converge at image point
• Can be projected on screen
• Formed by converging rays
• Can be inverted
• $d_i > 0$ (positive image distance)

Virtual images:

• Light rays appear to converge at image point (but don't)
• Cannot be projected on screen
• Formed by diverging rays (appear to come from)
• Always upright
• $d_i < 0$ (negative image distance)

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

Where:

• f = focal length
• $d_o$ = object distance (always positive)
• $d_i$ = image distance (+ for real, - for virtual)

Same form for thin lenses and spherical mirrors.

$m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$

Where:

• m = magnification
• $h_i$ = image height
• $h_o$ = object height

Interpretation:

• |m| > 1: image larger than object
• |m| < 1: image image smaller than object
• m > 0: image upright
• m < 0: image inverted
• |m| = 1: same size