A charge of −33 µC is distributed uniformly throughout a spherical volume of radius 19.0 cm. Determine the electric field (in N/C) due to this charge at the following distances from the center of the sphere. (Enter the radial component of the electric field.)
(a)
8.0 cm
E =( ? N/C) r̂
(b)
12.0 cm
E =( ? N/C) r̂
(c)
37.0 cm
E =( ? N/C) r̂
(a) is a trick question seeing if you are into Gauss' law:
inside a sphere:
The electric field E times the area of the sphere equals the charge INSIDE the sphere / eo
so
4 pi r^2 E = q / eo
but what is q INSIDE r = 8.0 ?
well the whole charge Q = -33 µC
and the whole volume of the sphere = (4/3) pi (19*10^-2)^2
so
charge / unit volume = -33 µC / [ (4/3) pi (19*10^-2)^3 ]
so our charge Qin =
{ -33 µC / [ (4/3) pi (19*10^-2)^3 ] }*[ (4/3) pi (8 *10^-2)^3 ]
= -33 µC * { 8^3 /19^3}
Now you can do the problem at radius = 8 cm
4 pi r^2 E = Qin / eo
4 pi (8*10^-2)^2 E = -33 µC * { 8^3 /19^3} / eo
by the way
eo is about 8.85*10^-12 C^2 /(N m^2)
part b is the same with new radius, 12 not 8
in fact the charge inside your sphere goes up with r^3 but the area goes up with r^2 so in there the field is proportional to r
12/8 = 1.5
so answer to b = 1.5 * answer to a
part c use the whole charge -33 uC but the radius = 0.37 meters
(a) Well, let's calculate the electric field at a distance of 8.0 cm from the center of the sphere. Now, as the charge is uniformly distributed throughout the sphere, the electric field inside will be the same as the electric field outside the sphere. So, we can apply the principles of Gauss's law to determine the electric field.
The formula to calculate the electric field due to a uniformly charged sphere is given by:
E = (kQ) / (r^2)
Where E is the electric field magnitude, k is Coulomb's constant (8.99 x 10^9 N m^2 / C^2), Q is the charge, and r is the distance from the center of the sphere.
In this case, the charge is -33 µC (negative because it's an electron-like charge) and the distance is 8.0 cm. So, plugging those values into the formula, we get:
E = (8.99 x 10^9 N m^2 / C^2) * (-33 x 10^-6 C) / (0.08 m)^2
Calculating this gives us the electric field at 8.0 cm from the center of the sphere. I'll let you do the math and find out the answer. Just remember, it will be in the form of N/C.
(b) For a distance of 12.0 cm, you can use the same formula as in part (a), but this time the distance will be different. Plug in the values and calculate the electric field. Again, it will be in N/C. Take a shot at it!
(c) Finally, for a distance of 37.0 cm, you can once again use the same formula. Calculate the electric field by substituting the values into the equation. And voila! You'll have the electric field at 37.0 cm. N/C, of course.
Remember, don't forget to put the answers in the r̂ notation!
To find the electric field due to the charge distributed uniformly throughout a spherical volume, you can use Gauss's Law. Gauss's Law states that the net electric flux through a closed surface is proportional to the net charge enclosed by the surface.
For a uniformly charged sphere, the electric field inside the sphere is zero. So, we only need to consider the electric field outside the sphere.
To determine the electric field at a given distance r from the center of the sphere, we can use the following equation:
E = (k * Q) / r^2
Where:
E is the electric field
k is Coulomb's constant (9.0 x 10^9 N*m^2/C^2)
Q is the total charge enclosed by the sphere
r is the distance from the center of the sphere
(a) Distance: 8.0 cm
Convert the distance to meters: 8.0 cm = 0.08 m
For a sphere, the total charge enclosed is the same as the total charge of the sphere, which is -33 µC = -33 x 10^-6 C.
Now, substitute the values into the formula:
E = (9.0 x 10^9 N*m^2/C^2 * (-33 x 10^-6 C)) / (0.08 m)^2
Simplifying the equation:
E = (-2.475 N/C) r̂
Therefore, at 8.0 cm from the center of the sphere, the electric field is -2.475 N/C in the radial direction.
(b) Distance: 12.0 cm
Convert the distance to meters: 12.0 cm = 0.12 m
Now, substitute the values into the formula:
E = (9.0 x 10^9 N*m^2/C^2 * (-33 x 10^-6 C)) / (0.12 m)^2
Simplifying the equation:
E = (-0.925 N/C) r̂
Therefore, at 12.0 cm from the center of the sphere, the electric field is -0.925 N/C in the radial direction.
(c) Distance: 37.0 cm
Convert the distance to meters: 37.0 cm = 0.37 m
Now, substitute the values into the formula:
E = (9.0 x 10^9 N*m^2/C^2 * (-33 x 10^-6 C)) / (0.37 m)^2
Simplifying the equation:
E = (-0.051 N/C) r̂
Therefore, at 37.0 cm from the center of the sphere, the electric field is -0.051 N/C in the radial direction.
To determine the electric field due to a charge distributed uniformly throughout a spherical volume at different distances from the center of the sphere, we can use Gauss's Law. Gauss's Law states that the electric field through a closed surface is proportional to the total charge enclosed by that surface.
To solve this problem, we need to calculate the charge enclosed by the surface at each distance and then calculate the electric field using the appropriate formula.
(a) 8.0 cm:
At 8.0 cm from the center of the sphere, the radius is 8.0 cm or 0.08 m. To calculate the charge enclosed by a sphere of radius 0.08 m, we need to determine the fraction of the total charge that is within that radius.
The volume of the sphere with radius 0.08 m is given by:
V = (4/3)πr^3 = (4/3)π(0.08)^3 = 0.00102 m^3
The charge enclosed by the sphere is obtained by multiplying the volume by the charge density:
Q = (0.00102 m^3) × (-33 μC/m^3)
Now we can use Gauss's Law to calculate the electric field:
E = Q / (4πε₀r^2)
Substituting the values, we get:
E = Q / (4πε₀r^2) = (-33 μC/m^3) × 0.00102 m^3 / (4πε₀ × (0.08 m)^2)
(b) 12.0 cm:
At 12.0 cm from the center of the sphere, the radius is 12.0 cm or 0.12 m. Following the same steps as before, we can calculate the charge enclosed by this radius and use Gauss's Law to determine the electric field at this distance.
(c) 37.0 cm:
At 37.0 cm from the center of the sphere, the radius is 37.0 cm or 0.37 m. Again, calculate the charge enclosed by this radius and use Gauss's Law to determine the electric field at this distance.
By following these steps, you can find the electric field at each distance from the center of the sphere.