The ionisation potential of mercury is 10.39 V. How far an electron must travel in an electric field of `1.5 xx 10^(6) V//m` to gain sufficient energy to ionise mercury ?

Options

A. `(10.39)/(1.6xx10^(-19))` m
B. `(10.39)/(2xx1.6xx10^(-19))` m
C. `10.39 xx 1.6 xx 10^(-19)` m
D. `(10.39)/(1.5xx10^(6))` m

Answer

To find out how far an electron must travel in an electric field to gain sufficient energy to ionize mercury, we can follow these steps:

### Step 1: Understand the Ionization Potential
The ionization potential (IP) of mercury is given as 10.39 V. This means that to ionize a mercury atom, an electron must gain at least 10.39 joules of energy per coulomb of charge.

### Step 2: Calculate the Work Required
The work done (W) to ionize mercury can be expressed in terms of the ionization potential:
\[ W = e \times \text{IP} \]
where \( e \) is the charge of an electron, approximately \( 1.6 \times 10^{-19} \) C.

### Step 3: Calculate the Work Done
Substituting the values:
\[ W = (1.6 \times 10^{-19} \, \text{C}) \times (10.39 \, \text{V}) \]
\[ W = 1.66 \times 10^{-18} \, \text{J} \]

### Step 4: Relate Work Done to Electric Field
The work done by the electric field (E) when an electron moves a distance \( d \) in an electric field \( E \) is given by:
\[ W = e \times E \times d \]

### Step 5: Set the Two Expressions for Work Equal
Since the work done to ionize mercury must equal the work done by the electric field, we can set the two expressions equal to each other:
\[ e \times \text{IP} = e \times E \times d \]

### Step 6: Cancel the Charge of the Electron
We can cancel \( e \) from both sides (as long as \( e \neq 0 \)):
\[ \text{IP} = E \times d \]

### Step 7: Solve for Distance \( d \)
Rearranging the equation to solve for \( d \):
\[ d = \frac{\text{IP}}{E} \]

### Step 8: Substitute the Values
Now substitute the known values:
- \( \text{IP} = 10.39 \, \text{V} \)
- \( E = 1.5 \times 10^6 \, \text{V/m} \)

So,
\[ d = \frac{10.39 \, \text{V}}{1.5 \times 10^6 \, \text{V/m}} \]

### Step 9: Calculate \( d \)
Calculating the distance:
\[ d = \frac{10.39}{1.5 \times 10^6} \]
\[ d \approx 6.93 \times 10^{-6} \, \text{m} \]
\[ d \approx 6.93 \, \mu\text{m} \]

### Final Answer
The distance the electron must travel in the electric field to gain sufficient energy to ionize mercury is approximately \( 6.93 \, \mu\text{m} \).

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