Askiitians Tutor Team
Last Activity: 5 Months ago
To estimate the upper limit of the error in calculating the force of interaction between charged spherical conductors using Coulomb's law, we need to consider the assumptions and approximations involved in the application of this law. Coulomb's law states that the force \( F \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:
F = k \frac{q_1 q_2}{r^2}
Here, \( k \) is Coulomb's constant. However, when dealing with spherical conductors, the situation is slightly more complex due to the distribution of charge over the surface of the spheres and their finite size.
Understanding the Geometry
For two charged spherical conductors with radii \( R \) and separated by a distance \( r \), the effective distance between their centers is not simply \( r \). Instead, we need to account for the radii of the spheres. The distance that we should consider in Coulomb's law is actually:
d = r - R_1 - R_2
where \( R_1 \) and \( R_2 \) are the radii of the two spheres. If we denote both spheres as having the same radius \( R \), the equation simplifies to:
d = r - 2R
Estimating the Error
The error in using Coulomb's law directly without adjusting for the finite size of the spheres can be significant, especially when the distance \( r \) is comparable to the radii \( R \). The upper limit of the error can be estimated by considering the difference between the actual force and the force calculated using the point charge approximation.
- The actual force \( F_{actual} \) can be expressed as:
F_{actual} = k \frac{q_1 q_2}{(r - 2R)^2}
- The force calculated using the point charge approximation is:
F_{point} = k \frac{q_1 q_2}{r^2}
The error \( E \) can then be defined as:
E = F_{point} - F_{actual}
Calculating the Error Bound
To find the upper limit of the error, we can analyze the ratio of the two forces:
\(\frac{F_{point}}{F_{actual}} = \frac{(r - 2R)^2}{r^2}\)
As \( R \) approaches \( r/2 \), the ratio approaches zero, indicating that the error becomes significant. Thus, we can express the upper limit of the error in terms of the ratio of the distances:
E \approx F_{point} \left(1 - \left(\frac{r - 2R}{r}\right)^2\right)
In practical terms, if \( R \) is much smaller than \( r \), the error is minimal. However, if \( R \) is not negligible compared to \( r \), the error can be substantial, and adjustments must be made to account for the finite size of the spheres.
Conclusion
In summary, when calculating the force of interaction between charged spherical conductors, one must consider the effective distance between their centers, which includes their radii. The upper limit of the error can be significant if the spheres are large relative to the distance separating them. Thus, careful consideration of geometry and charge distribution is essential for accurate calculations.
