If the kinetic energy of an electron is increased 4 times, the wavelength of the Broglie wave associated with it would become:
Understand the Problem
The question is asking how the wavelength of a de Broglie wave associated with an electron changes if its kinetic energy is increased four times. The problem involves understanding the relationship between kinetic energy and wavelength, where an increase in energy affects the wavelength inversely.
Answer
The new wavelength is \( \frac{1}{2} \lambda \).
Answer for screen readers
The new wavelength of the de Broglie wave would become ( \frac{1}{2} \lambda ).
Steps to Solve
- Understanding Kinetic Energy and Momentum Relationship
Kinetic energy (KE) is given by the formula: $$ KE = \frac{1}{2} mv^2 $$ where ( m ) is the mass and ( v ) is the velocity of the electron.
- Finding Momentum in terms of Kinetic Energy
Momentum ( p ) is related to kinetic energy through the equation: $$ p = \sqrt{2m \cdot KE} $$ Substituting the expression for kinetic energy into the momentum equation: $$ p = \sqrt{2m \cdot \left(\frac{1}{2} mv^2\right)} = mv $$
- De Broglie Wavelength Expression
The de Broglie wavelength ( \lambda ) is expressed as: $$ \lambda = \frac{h}{p} $$ where ( h ) is Planck's constant.
- Analyzing Change in Kinetic Energy
If the kinetic energy is increased four times, we denote the new kinetic energy as: $$ KE' = 4 \cdot KE $$
- Determining New Momentum After Energy Increase
Now substituting ( KE' ) into the momentum formula: $$ p' = \sqrt{2m(4KE)} = \sqrt{8m \cdot KE} = 2\sqrt{2m \cdot KE} = 2p $$
- Calculating New Wavelength
Substituting ( p' = 2p ) into the de Broglie wavelength formula: $$ \lambda' = \frac{h}{p'} = \frac{h}{2p} = \frac{1}{2} \lambda $$
The new wavelength of the de Broglie wave would become ( \frac{1}{2} \lambda ).
More Information
When the kinetic energy of an electron is increased, its momentum increases as well. Since wavelength is inversely proportional to momentum, increasing kinetic energy results in a shorter wavelength.
Tips
- Confusing kinetic energy with momentum; it's important to note that they are directly related but are not the same thing.
- Miscalculating the relationship between kinetic energy and wavelength, specifically overlooking the square root when relating momentum and kinetic energy.
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