An electron beam is acceleration by a potential difference V to hit a metallic target to produce X-rays. It produces continuous as well as characteristic X-rays. If `lambda_(min)` is the smallest possible wavelength of X-rays in the spectrum, the variation of `log lambda_(min)` with `log v` is correctly represented in

Options

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Explanation

To solve the problem of how the minimum wavelength of X-rays (\( \lambda_{min} \)) varies with the accelerating potential difference (\( V \)), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy Relationship**:
When an electron is accelerated through a potential difference \( V \), it gains kinetic energy equal to the work done on it by the electric field. This energy is given by:
\[ E = eV \]
where \( e \) is the charge of the electron.

2. **Relate Energy to Wavelength**:
The energy of a photon (which corresponds to the X-rays produced) is related to its wavelength by the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light. For the minimum wavelength (\( \lambda_{min} \)), we can equate the two expressions for energy:
\[ eV = \frac{hc}{\lambda_{min}} \]

3. **Rearranging for \( \lambda_{min} \)**:
Rearranging the equation gives us:
\[ \lambda_{min} = \frac{hc}{eV} \]

4. **Taking Logarithms**:
To analyze the relationship in logarithmic terms, we take the logarithm of both sides:
\[ \log(\lambda_{min}) = \log\left(\frac{hc}{eV}\right) \]

5. **Using Logarithmic Properties**:
Applying the properties of logarithms, we can separate the terms:
\[ \log(\lambda_{min}) = \log(hc) - \log(e) - \log(V) \]
This can be rearranged to:
\[ \log(\lambda_{min}) = \log(hc/e) - \log(V) \]

6. **Identifying the Linear Relationship**:
We can express this in the form of a linear equation:
\[ \log(\lambda_{min}) = -\log(V) + \log(hc/e) \]
Here, if we let \( y = \log(\lambda_{min}) \) and \( x = \log(V) \), we see that this is a linear equation of the form:
\[ y = mx + c \]
where the slope \( m = -1 \) and the y-intercept \( c = \log(hc/e) \).

7. **Graph Interpretation**:
Since the slope is negative, this indicates that as \( V \) increases, \( \lambda_{min} \) decreases. The graph of \( \log(\lambda_{min}) \) versus \( \log(V) \) will be a straight line with a negative slope.

### Conclusion:
The relationship between \( \log(\lambda_{min}) \) and \( \log(V) \) is linear with a negative slope of -1. Therefore, the correct representation of the variation of \( \log(\lambda_{min}) \) with \( \log(V) \) is a straight line with a negative slope.