explanation of uniform continuity real analysis
Answers
Answer:
t is given that the energy of the electron beam used to bombard gaseous hydrogen at room temperature is 12.5 eV. Also, the energy of the gaseous hydrogen in its ground state at room temperature is 13.6 eV.
When gaseous hydrogen is bombarded with an electron beam, the energy of the gaseous hydrogen becomes 13.6 + 12.5 eV i.e., 1.1 eV.
Orbital energy is related to orbit level (n)as:
Forn= 3,
This energy is approximately equal to the energy of gaseous hydrogen. It can be concluded that the electron has jumped fromn= 1 ton= 3 level.
During its de-excitation, the electrons can jump fromn= 3 ton= 1 directly, which forms a line of the Lyman series of the hydrogen spectrum.
We have the relation for wave number for Lyman series as:
Where,
Ry= Rydberg constant = 1.097 × 107m1
λ=Wavelength of radiation emitted by the transition of the electron
Forn= 3, we can obtainλas:
If the electron jumps fromn= 2 ton= 1, then the wavelength of the radiation is given as:
If the transition takes place from n = 3 to n = 2, then the wavelength of the radiation is given as:
This radiation corresponds to the Balmer series of the hydrogen spectrum.
Hence, in Lyman series, two wavelengths i.e., 102.5 nm and 121.5 nm are emitted. And in the Balmer series, one wavelength i.e., 656.33 nm is emitted.
Physics Syllabus
General:Units and dimensions, dimensional analysis; least count, significant figures; Methods of measurement and error analysis for physical quantities pertaining to the following experiments: Experiments based on using Vernier calipers and screw gauge (micrometer), Determination of g using simple pendulum, Young's modulus by Searle's method, Specific heat of a liquid using calorimeter, focal length of a concave mirror and a convex lens using u-v method, Speed of sound using resonance column, Verification of Ohm's law using voltmeter and ammeter, and specific resistance of the material of a wire using meter bridge and post office box.
Mechanics:Kinematics in one and two dimensions (Cartesian coordinates only), projectiles; Uniform Circular motion; Relative velocity.
Step-by-step explanation:
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In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.
Continuous functions can fail to be uniformly continuous if they are unbounded on a finite domain, such as {\displaystyle f(x)={\tfrac {1}{x}}} {\displaystyle f(x)={\tfrac {1}{x}}} on (0,1), or if their slopes become unbounded on an infinite domain, such as {\displaystyle f(x)=x^{2}} f(x)=x^{2} on the real line. However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map) .
Although ordinary continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space.
