Force between two point charges \( q_1 \) and \( q_2 \) placed in vacuum at \( r \) cm apart is \( F \). Force between them when placed in a medium having dielectric \( K = 5 \) at \( r/5 \) cm apart will be:
- \( \frac{F}{25} \)
- \( 5F \)
- \( \frac{F}{5} \)
- \( 25F \)
The Correct Option is B
Approach Solution - 1
To find the force between two point charges in different conditions, we use Coulomb's Law, which states that the electrostatic force \(F\) between two point charges \( q_1 \) and \( q_2 \) in vacuum separated by a distance \( r \) is given by:
\(F = \frac{{k \cdot q_1 \cdot q_2}}{{r^2}}\)
where \( k \) is Coulomb's constant.
When the charges are placed in a medium with dielectric constant \( K \), the force is reduced by the factor \( K \). The modified force \( F_{\text{medium}} \) is given by:
\(F_{\text{medium}} = \frac{{F}}{{K}}\)
Now, we are to find the force when the charges are placed in a medium with dielectric constant \( K = 5 \) and at a new distance \( r' = \frac{r}{5} \).
- First, adjust the force for the dielectric medium:
\(F_{\text{medium}} = \frac{F}{5}\)
- Next, adjust for the new distance. The force between two charges is inversely proportional to the square of the distance between them:
Using the new distance, \( r' = \frac{r}{5} \), the force becomes:
\(F_{\text{new}} = \frac{k \cdot q_1 \cdot q_2}{\left(\frac{r}{5}\right)^2} = \frac{k \cdot q_1 \cdot q_2}{\frac{r^2}{25}} = 25 \cdot F\)
- Combine both effects:
\(F_{\text{final}} = \frac{F_{\text{new}}}{K} = \frac{25F}{5} = 5F\)
Thus, when the charges are placed in a medium with dielectric constant \( K = 5 \) and distance \( \frac{r}{5} \), the force between them is \( 5F \).
The correct option is: \( 5F \)
Approach Solution -2
The force between two point charges \(q_1\) and \(q_2\) separated by a distance \(r\) in a vacuum is given by Coulomb’s law:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2}, \]
where \(\epsilon_0\) is the permittivity of free space.
In a medium with dielectric constant \(K\), the permittivity changes from \(\epsilon_0\) to \(K \epsilon_0\). This reduces the effective force between the charges by a factor of \(K\). Thus, the force in the medium, if the distance remained \(r\), would be:
\[ F_{\text{medium}} = \frac{1}{4 \pi K \epsilon_0} \frac{q_1 q_2}{r^2} = \frac{F}{K}. \]
For \(K = 5\), this becomes:
\[ F_{\text{medium}} = \frac{F}{5}. \]
Now, since the distance between the charges is reduced to \(\frac{r}{5}\), we need to adjust for this change. Coulomb’s force varies inversely with the square of the distance, so reducing the distance by a factor of \(\frac{1}{5}\) increases the force by a factor of \(\left(\frac{1}{5}\right)^{-2} = 25\).
Combining both effects (the dielectric and the reduced distance), the modified force \(F'\) in the medium is:
\[ F' = \frac{F}{5} \times 25 = 5F. \]
Thus, the force between the charges in the medium, with the distance changed to \(\frac{r}{5}\), is increased by a factor of 5 compared to the original force in a vacuum. Therefore, the answer is: \[ 5F. \]
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