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^Problem solving tips

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^Problem solving tips

  • If a particle is moving on a circular path with angular speed w about the center, then its angular speed about a point on the opposite side of the diameter passing through it is w/2.
  • If uniform speed remains constant then the motion is called uniform circular motion. Magnitude of v, at & an respectively are
  • On a rigid body all points undergo same angular displacement (θ) in same time interval thus all the points on a rigid body rotate at same angular speed (ω) & have angular acceleration (α), thus v, at & an ∝ r. Thus a point situated farthest from center circle will have maximum value of speed, tangential acceleration & normal acceleration. Also a point at center will have zero values of v, at & an.
  • is a unit vector directed radially or normally outwards.

^For a body just released from a height

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^For a body just released from a height

If a body is dropped from a height h,

1. it reaches the ground with speed [using v – s relation]

2. The time taken by it to reach the ground is [using s – t relation]

 

^Crossing a river

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^Crossing a river

Suppose a person start rowing his boat from point O with velocity vbr (w.r.t. river i.e. without considering the effect of water) towards point S in order to cross a river of width ‘w’ flowing at a speed ‘vr’ towards x axis. Let under the effect of river current boat has net velocity the boat vb (w.r.t. ground i.e. resultant vbr & vr) & he reaches point R, then the distance SR covered downstream (i.e. distance along the stream) is called drift.

^Brewster’s law

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^Brewster’s law

Brewster discovered that when ordinary light is incident on the surface of a transparent medium the reflected light is partially plane polarized.

The extent of polarization depends on the angle of incidence. For a particular angle of incidence (called polarizing or Brewster angle (ip or p)

1. the reflected light is found to be completely polarized with its vibrations perpendicular to the plane of incidence.

2. the reflected & transmitted rays are perpendicular to each other, as shown in figure.

3. μ = tan p                           [called Brewster law

At i = p Ray 1: Incident ray (unpolarised)

Ray 2: Reflected ray (completely polarised)

Ray 3: Refracted ray (partially polarised)

For a ray incident form air to glass

aμg = 3/2, so p = tan-1  3/2  ⇒ p = 56.30

For a ray incident form air to water

aμw = 4/3,  so p = tan-1  4/3 ⇒  p = 53.10

^Malus law

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^Malus law

Let I0 be the intensity and ‘A’ the amplitude of the unpolarized light falling on the polarizer & θ be the angle between the pass axis of the polaroid & amplitude vector, then the component Acosθ, parallel to the pass axis (y) of polarizer passes through it & the component Asinθ, perpendicular to pass axis is blocked by the polarizer, thus the intensity of light transmitted by the polarizer is

I = k (A cosθ)2 = k A2 cos2 θ

or         I = I0 cos2 θ        [called Malus law

Actually the phase angle for a beam of unpolarised light can have any value from 00 to 3600. Now as the average value of cos2 θ for the range 00 to 3600 is 0.5, thus the average intensity of light transmitted by a polarizer is only 0.5 I0.  Following observations can be explained on the basis of Malus law.

1. Two polarizes with parallel pass axis (y & y)

2. Two polarizes with crossed pass axis (y & z)

^Light through a polarizer

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^Light through a polarizer

Let ordinary light propagating along +x axis falls normally on a tourmaline crystal having pass axis parallel to y axis as shown in the following diagram.  On passing through the tourmaline crystal the electric field vibration parallel to pass axis are transmitted while the electric field vibration normal to pass axis are absorbed by the tourmaline.

^How do polarisers work

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^How do polarisers work 

A Polaroid is a thin commercial sheet containing long chain of hydrocarbons which become conducting at optical frequencies & absorb the electric field parallel to chains & transmit the electric field perpendicular to the chain length. This is why a direction perpendicular to the chain length is called the transmission axis or pass axis of the polarizer. For shown drawing y axis is pass axis.

^Polarisers

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^Polarisers

Any device or material that plane polarizes the unpolarised light passed through it is called a polarizer.  Nicol prism, Tourmaline crystal, Polaroid etc. are some commonly used Polarisers.

^Plane polarized light

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^Plane polarized light

If the electric field vector of a light wave vibrates just in one direction perpendicular to the direction of wave propagation, then it is said to be linearly polarized along that direction. Human eyes can’t distinguish between an unpolarised light and a polarized light, but the eyes of a bee can.

^Ordinary or unpolarized light

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^Ordinary or unpolarized light

Light is made up of the superposition of a huge number of light waves emitted by individual atoms whose electric fields are in random directions. As a result the direction of vibration (DOV) of electric field vector at any given point of space keeps on changing very rapidly & randomly but always remain perpendicular to the direction of propagation (DOP). Such a light is called randomly polarized or simply unpolarized light. If a light wave propagates along x axis then tip of the electric field vector traces an irregular planar curve & there can be infinite directions of vibrations of electric field vector in the yz plane as shown in the diagram.

Light from a light bulb, candle, Sun is unpolarized. Usually we represent electric field vibrations in a ordinary light by dots & double arrows. Here double arrows represent the vibrations in the plane of paper (let xy plane) i.e. along ± y – axis and small dots represent vibrations perpendicular to the plane of paper along i.e. ± z – axis.

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