The energy, E, of a photon can be expressed as E = hf where f is the frequency and h is Planck’s constant. The dimensions of h are the same as that of
- (a)linear momentum
- (b)angular momentum
- (c)displacement
- (d)torque
Correct — B, angular momentum. From E = hf, h = E / f, which has units of energy x time = joule-second (J.s). Angular momentum (L = I x omega, or mvr) also has units kg.m^2.s^-1 = J.s, that is the same dimensional formula [M L^2 T^-1]. So Planck's constant is dimensionally identical to angular momentum — both are quantities of 'action'.
- (a)linear momentum — Linear momentum p = mv has units kg.m.s^-1, dimensional formula [M L T^-1] — one power of length short of Planck's constant, which is [M L^2 T^-1].
- (c)displacement — Displacement is just a length, [L]. It shares none of the mass or time dependence of Planck's constant, so it cannot match.
- (d)torque — Torque = force x distance = N.m = kg.m^2.s^-2, dimensional formula [M L^2 T^-2] — the same as ENERGY, not as h. Planck's constant is energy x TIME, one extra power of time (T^-1, not T^-2).
Planck's constant h (about 6.626 x 10^-34 J.s) links a photon's energy to its frequency through E = hf. Dividing energy (joule) by frequency (per second) gives the joule-second. Quantities with units of energy x time — equivalently momentum x distance — are called 'action', and angular momentum shares exactly these dimensions [M L^2 T^-1].
Do the dimensional bookkeeping straight from E = hf: h = E / f = energy / (1/time) = energy x time. Now test the options — linear momentum is [M L T^-1], displacement is [L], and torque equals energy at [M L^2 T^-2]; none of these match energy x time. Only angular momentum, [M L^2 T^-1], equals energy x time, so it is the answer. This is also why the reduced constant h-bar = h / 2pi turns up wherever angular momentum is quantised.
- h is about 6.626 x 10^-34 J.s; the joule-second is the unit of 'action'.
- Dimensional formula of h is [M L^2 T^-1] — the same as angular momentum.
- Linear momentum is [M L T^-1] and torque/energy is [M L^2 T^-2] — neither matches h.
- Angular momentum is quantised in units of h-bar = h / 2pi (for example Bohr's condition L = n x h-bar).
- Picking torque because it is [M L^2 T^-2] like energy — but h is energy x TIME, one extra power of time.
- Confusing linear momentum [M L T^-1] with angular momentum [M L^2 T^-1].
Dimensions are asked as 'which quantity has the same dimensions as X' — reduce X to base units (M, L, T) and match.
Assertion (A): In the visible spectrum of light, red light is more energetic than green light. Reason (R): The wavelength of red light is more than that of green light.
- (a) Both A and R are individually true and R is the correct explanation of A
- (b) Both A and R are individually true but R is not the correct explanation of A
- (c) A is true but R is false
- (d) A is false but R is true
Answer(d) A is false but R is true
Same Planck relation E = hf (equivalently E = hc/lambda). There the point is that longer-wavelength red light is LESS energetic than green, so the assertion is false; here the task is to see that h = E/f carries the dimensions of angular momentum. Both rest on E = hf.
- practice — not a real PYQ
Which one of the following pairs has the same dimensional formula?
- (a)Work and torque
- (b)Force and momentum
- (c)Pressure and energy
- (d)Power and force
Answer(a) Work and torque — both are [M L^2 T^-2].
- practice — not a real PYQ
The dimensional formula of Planck's constant is
- (a)[M L^2 T^-1]
- (b)[M L T^-1]
- (c)[M L^2 T^-2]
- (d)[M L^-1 T^-1]
Answer(a) [M L^2 T^-1] — energy x time, the same as angular momentum.