A star which is emitting radiation at a wavelength of 5000A is approaching the earth with a velocity of `1.50 xx10^(6) m//s` The change in wavelegth of the radiation as received on the earth is

Options

A. 0.25 Å
B. 2.5 Å
C. 25 Å
D. 250 Å

Answer

To solve the problem of determining the change in wavelength of radiation emitted by a star approaching Earth, we can use the Doppler effect for light. Here's a step-by-step solution:

### Step 1: Understand the Doppler Effect for Light
The Doppler effect describes how the observed frequency (and thus wavelength) of a wave changes when the source of the wave is moving relative to an observer. For light waves, when the source is moving towards the observer, the observed wavelength decreases.

### Step 2: Identify Given Values
- Emitted wavelength (\( \lambda_0 \)): 5000 Å (angstroms)
- Velocity of the star (\( v \)): \( 1.50 \times 10^6 \, \text{m/s} \)
- Speed of light (\( c \)): \( 3.00 \times 10^8 \, \text{m/s} \)

### Step 3: Use the Doppler Effect Formula
The formula for the observed wavelength (\( \lambda \)) when the source is moving towards the observer is given by:

\[ \lambda = \lambda_0 \left(1 - \frac{v}{c}\right) \]

### Step 4: Substitute the Values
Substituting the known values into the formula:

\[ \lambda = 5000 \, \text{Å} \left(1 - \frac{1.50 \times 10^6}{3.00 \times 10^8}\right) \]

### Step 5: Calculate the Fraction
Calculate \( \frac{v}{c} \):

\[ \frac{1.50 \times 10^6}{3.00 \times 10^8} = 0.005 \]

### Step 6: Substitute Back into the Wavelength Equation
Now substitute this back into the equation for \( \lambda \):

\[ \lambda = 5000 \, \text{Å} \left(1 - 0.005\right) \]
\[ \lambda = 5000 \, \text{Å} \times 0.995 \]
\[ \lambda = 4975 \, \text{Å} \]

### Step 7: Determine the Change in Wavelength
The change in wavelength (\( \Delta \lambda \)) can be calculated as:

\[ \Delta \lambda = \lambda_0 - \lambda = 5000 \, \text{Å} - 4975 \, \text{Å} = 25 \, \text{Å} \]

### Final Answer
The change in wavelength of the radiation as received on Earth is \( 25 \, \text{Å} \).

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