In a photoelectric emission , from a black surface, threshold frequency is `v_0` . For two incident radiations of frequencies `v_1 and v_2` , the maximum values of kinetic energy of the photoelectrons emitted in the two cases is in the ration 1,P . If `pv_1 -v_2 =2v_0` , find the value of |p|.
`{:(0,1,2,3,4,5,6,7,8,9):}`

Explanation

To solve the problem step by step, we will use the principles of photoelectric emission and the equations derived from Einstein's photoelectric equation.

### Step 1: Understand the Given Information
We are given:
- Threshold frequency: \( v_0 \)
- Frequencies of incident radiations: \( v_1 \) and \( v_2 \)
- The ratio of maximum kinetic energies of emitted photoelectrons: \( \frac{K_1}{K_2} = \frac{1}{p} \)
- The equation relating \( p \), \( v_1 \), and \( v_2 \): \( p v_1 - v_2 = 2 v_0 \)

### Step 2: Write the Photoelectric Equation
According to the photoelectric effect, the kinetic energy (\( K \)) of the emitted photoelectrons can be expressed as:
\[ K = h\nu - h\nu_0 \]
Where \( h \) is Planck's constant, \( \nu \) is the frequency of the incident radiation, and \( \nu_0 \) is the threshold frequency.

### Step 3: Write the Kinetic Energy Equations
For the two frequencies \( v_1 \) and \( v_2 \), we can write:
1. For \( v_1 \):
\[ K_1 = h v_1 - h v_0 \]
2. For \( v_2 \):
\[ K_2 = h v_2 - h v_0 \]

### Step 4: Relate the Kinetic Energies
From the ratio of the kinetic energies:
\[ \frac{K_1}{K_2} = \frac{1}{p} \]
This implies:
\[ \frac{h v_1 - h v_0}{h v_2 - h v_0} = \frac{1}{p} \]
Cancelling \( h \) from both sides gives:
\[ \frac{v_1 - v_0}{v_2 - v_0} = \frac{1}{p} \]

### Step 5: Rearranging the Equation
Cross-multiplying gives:
\[ v_1 - v_0 = \frac{1}{p}(v_2 - v_0) \]
Multiplying both sides by \( p \):
\[ p(v_1 - v_0) = v_2 - v_0 \]
Rearranging gives:
\[ p v_1 - v_2 = p v_0 - v_0 \]
This can be simplified to:
\[ p v_1 - v_2 = (p - 1)v_0 \]

### Step 6: Set the Two Equations Equal
We have two expressions for \( p v_1 - v_2 \):
1. From the kinetic energy ratio: \( p v_1 - v_2 = (p - 1)v_0 \)
2. From the given equation: \( p v_1 - v_2 = 2 v_0 \)

Setting these equal gives:
\[ (p - 1)v_0 = 2 v_0 \]

### Step 7: Solve for \( p \)
Dividing both sides by \( v_0 \) (assuming \( v_0 \neq 0 \)):
\[ p - 1 = 2 \]
Thus:
\[ p = 3 \]

### Final Answer
The value of \( |p| \) is \( 3 \).

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