Final answer:
The total charge of the solid insulating sphere can be found by applying Gauss's Law to calculate the charge enclosed within Gaussian surfaces at 10.0 cm and 50.0 cm from the center, utilizing the electric field values provided at these distances.
Step-by-step explanation:
To determine the total charge of the solid insulating sphere, we can use Gauss's Law, which relates the electric field to the charge enclosed within a Gaussian surface. Here, we know the electric field at two points: 3600 N/C at 10.0 cm from the center and 200 N/C at 50.0 cm from the center.
For the first point, since it's inside the solid sphere (radius 5.00 cm), the field would be due to the charge within a Gaussian sphere of radius 10.0 cm concentric with the solid sphere. By using the formula E = k * (Q_enclosed / r^2), where k is Coulomb's constant (approximately 8.99 x 10^9 N m^2/C^2), we can calculate the charge enclosed (Q_enclosed).
At 10.0 cm:
E = 3600 N/C,
k = 8.99 x 10^9 N m^2/C^2,
r = 10.0 cm = 0.1 m.
Solving for Q_enclosed, we get Q_enclosed = E * r^2 / k.
Q_enclosed = (3600 N/C) * (0.1 m)^2 / (8.99 x 10^9 N m^2/C^2).
This calculation gives us the charge within the radius of 10.0 cm. To find the total charge of the sphere, we repeat the similar calculation for the field at 50.0 cm, where the Gaussian surface now encompasses the entire charged sphere. This field is due to the entire sphere, not just a part of it. Since the field at 50.0 cm is due to the total charge on the sphere, we can calculate the total charge directly.
At 50.0 cm:
E = 200 N/C,
k = 8.99 x 10^9 N m^2/C^2,
r = 50.0 cm = 0.5 m.
Q_total = E * r^2 / k = (200 N/C) * (0.5 m)^2 / (8.99 x 10^9 N m^2/C^2).
By performing these calculations, we can determine the total charge of the solid insulating sphere. Please note that while the provided examples include different configurations, the principle behind using Gauss's Law remains the same for enclosed charges.