A hollow sphere has a uniform volume charge density of 4.61 nC/m3. The inner radius is a = 14.6 cm and the outer radius is b = 43.8 cm.
What is the magnitude of the electric field at 21.9 cm from the center of the sphere?
What is the magnitude of the electric field at 140. cm from the center of the sphere?
gauss law applies.
volume1=4/3 PI ((.219^3)-(.146)^3)
now to get the charge contained, that volume x 4.61nC/m^3
E=kQ/.219^2
second part, all the charge is contained
volumecharge=4/3 pi ((.438^2)-(.146)^2)
charge = volumecharge*chargedensity
= volumecharge*4.61nC/m^3
E=kQ/1.40^2
bobpursley did you intentionally square the radius values for the second volume, instead of making it to the power of 3?
" volumecharge=4/3 pi ((.438^2)-(.146)^2) "
To find the magnitude of the electric field at a given distance from the center of the sphere, we can use Gauss's Law. Gauss's Law states that the electric field at a point outside a charged sphere is the same as if all the charge were concentrated at the center of the sphere.
Let's solve these step-by-step:
Step 1: Calculate the total charge enclosed within the Gaussian surface at a distance of 21.9 cm from the center of the sphere.
The Gaussian surface is a sphere with a radius of 21.9 cm. The charge enclosed within this sphere is the charge density multiplied by the volume of the sphere. So we calculate:
V = (4/3)πr^3
V = (4/3)π(0.219 m)^3
V ≈ 0.0909 m^3
Q_enclosed = charge density * volume
Q_enclosed = (4.61 nC/m^3) * (0.0909 m^3)
Q_enclosed ≈ 0.4191 nC
Step 2: Calculate the electric field at a distance of 21.9 cm using Gauss's Law.
Gauss's Law states that the electric field at a distance r from the center of a sphere is given by:
E = (k * Q_enclosed) / r^2
where k is the Coulomb constant.
Substituting the values we obtained:
E = (9 x 10^9 N*m^2/C^2) * (0.4191 x 10^-9 C) / (0.219 m)^2
E ≈ 3.9669 x 10^4 N/C
Therefore, the magnitude of the electric field at a distance of 21.9 cm from the center of the sphere is approximately 3.9669 x 10^4 N/C.
Now let's calculate the electric field at a distance of 140 cm from the center of the sphere:
Step 1: Calculate the total charge enclosed within the Gaussian surface at a distance of 140 cm from the center of the sphere.
The Gaussian surface is a sphere with a radius of 140 cm. The charge enclosed within this sphere is the charge density multiplied by the volume of the sphere. So we calculate:
V = (4/3)πr^3
V = (4/3)π(1.40 m)^3
V ≈ 11.76 m^3
Q_enclosed = charge density * volume
Q_enclosed = (4.61 nC/m^3) * (11.76 m^3)
Q_enclosed ≈ 54.1296 nC
Step 2: Calculate the electric field at a distance of 140 cm using Gauss's Law.
Using Gauss's Law:
E = (k * Q_enclosed) / r^2
Substituting the values:
E = (9 x 10^9 N*m^2/C^2) * (54.1296 x 10^-9 C) / (1.40 m)^2
E ≈ 2.0717 x 10^3 N/C
Therefore, the magnitude of the electric field at a distance of 140 cm from the center of the sphere is approximately 2.0717 x 10^3 N/C.
To find the magnitude of the electric field at a given distance from the center of the sphere, we can use Gauss's law. Gauss's law relates the electric field to the enclosed charge within a closed surface.
First, we need to determine the total charge enclosed within the sphere. We know that the charge density is 4.61 nC/m^3, so to find the total charge, we multiply the charge density by the volume of the sphere.
The volume of a hollow sphere can be calculated using the formula:
V = (4/3) * π * (b^3 - a^3)
where a is the inner radius and b is the outer radius.
Let's calculate the volume of the hollow sphere:
V = (4/3) * π * [(0.438)^3 - (0.146)^3]
Next, we multiply the volume by the charge density to find the total charge enclosed within the sphere:
Total charge = Volume * Charge density
Now, we can calculate the electric field by applying Gauss's law. According to Gauss's law, the electric field is given by:
E = (1 / (4πε0)) * (Total charge / r^2)
where ε0 is the permittivity of free space and r is the distance from the center of the sphere.
Now, we substitute the values into the formula to calculate the electric field at various distances:
For r = 0.219 m (or 21.9 cm):
E = (1 / (4πε0)) * (Total charge / (0.219)^2)
For r = 1.40 m (or 140 cm):
E = (1 / (4πε0)) * (Total charge / (1.40)^2)
By plugging in the values and performing the calculations, we can find the magnitude of the electric field at those mentioned distances from the center of the sphere.