An insulating solid sphere of radius a = 0.5 m. If the sphere carries a total positive charge of 15 nC, calculate the magnitude of the electric field at r = 0.2 m from the center of the sphere.
15
answer
sorry the answer is 15
500
To calculate the magnitude of the electric field at a distance of 0.2 m from the center of the sphere, we can use Gauss's Law. Gauss's Law states that the electric flux through a closed surface is equal to the total enclosed charge divided by the permittivity of free space (ε₀).
The first step is to determine if the given situation satisfies the conditions for using Gauss's Law. Gauss's Law is most easily applied to situations with high symmetry, such as when the electric field is radially symmetric. In this case, since we have a solid sphere with a uniform charge distribution, the symmetry allows us to apply Gauss's Law.
The next step is to choose an appropriate Gaussian surface. A Gaussian surface can be any closed surface that encloses the charged sphere and has a high degree of symmetry. Since we have a spherical charge distribution, the most convenient choice is a spherical Gaussian surface centered on the charged sphere.
Now that we have our Gaussian surface, we need to calculate the electric flux through this surface. The electric flux is equal to the electric field (E) multiplied by the area (A) of the Gaussian surface.
Since the electric field is radially symmetric for a uniformly charged sphere, the magnitude of the electric field will be constant on the Gaussian surface and will only depend on the distance from the center. This allows us to factor the electric field out of the integral.
Thus, the electric flux can be calculated as the product of the electric field magnitude (E) and the area (A) of the Gaussian surface:
Φ = E * A
The electric flux (Φ) through the Gaussian surface is equal to the total enclosed charge (Q_enclosed) divided by the permittivity of free space (ε₀):
Φ = Q_enclosed / ε₀
Since we are given the total positive charge (Q_total = 15 nC) and we want to calculate the electric field at a distance of 0.2 m from the center of the sphere, we need to determine the charge enclosed within a Gaussian sphere of radius r = 0.2 m.
The charge enclosed within a Gaussian surface can be calculated as the fraction of the total charge that is enclosed within the surface. Since we have a uniformly charged sphere, the fraction of the total charge enclosed within a sphere of radius r is equal to the ratio of the volume of the enclosed sphere to the volume of the total sphere.
The volume of a sphere is given by the formula:
V = (4/3) * π * r^3
Therefore, the charge enclosed within a sphere of radius r is given by:
Q_enclosed = (4/3) * π * r^3 * density
Where density is the charge per unit volume, which is equal to the total charge divided by the total volume of the sphere.
The total volume of the sphere is given by:
V_total = (4/3) * π * a^3
Where 'a' is the radius of the sphere.
Finally, we can substitute the values into the equation to calculate the magnitude of the electric field at a distance of 0.2 m from the center of the sphere:
Q_enclosed = (4/3) * π * (0.2)^3 * (15 * 10^-9 C) / [(4/3) * π * (0.5)^3]
Q_enclosed = 0.0024 C
Now, we can calculate the electric field using Gauss's Law:
Φ = Q_enclosed / ε₀ = E * A
E = Φ / A
The area (A) of the Gaussian surface can be calculated as:
A = 4 * π * r^2
Plugging in the values:
E = (Q_enclosed / ε₀) / (4 * π * (0.2)^2)
E = (0.0024 C / (8.854 * 10^-12 C²/N·m²)) / (4 * π * (0.2)^2)
Calculating this expression will give us the magnitude of the electric field at a distance of 0.2 m from the center of the sphere.