A metal sphere of radius 5.00 cm is initially uncharged. How many electrons would have to be placed on the sphere to produce an electric field of magnitude 1.59 105 N/C at a point 7.98 cm from the center of the sphere?
To find the number of electrons required to produce the given electric field, we need to use the concept of electric field due to a point charge and the relationship between charge and number of electrons.
The electric field due to a point charge is given by the equation:
E = k * q / r^2
Where:
E is the electric field magnitude,
k is the Coulomb's constant (8.988 × 10^9 N m^2/C^2),
q is the charge magnitude, and
r is the distance from the point charge.
In this case, we want to find the number of electrons required to produce an electric field of magnitude E = 1.59 × 10^5 N/C at a distance r = 7.98 cm from the center of the sphere.
We can rearrange the equation to solve for the charge magnitude q:
q = E * r^2 / k
Now, let's calculate the charge magnitude required:
q = (1.59 × 10^5 N/C) * (7.98 × 10^(-2) m)^2 / (8.988 × 10^9 N m^2/C^2)
Simplifying the equation, we have:
q = 1.27203 × 10^(-13) C
To determine the number of electrons, we need to use the elementary charge e, which is the charge of an electron:
e = 1.602 × 10^(-19) C
The number of electrons (N) required to produce the given charge can be calculated by dividing q by e:
N = q / e
Substituting the values, we get:
N = (1.27203 × 10^(-13) C) / (1.602 × 10^(-19) C)
Calculating further, we find:
N ≈ 793,446,613 electrons
Therefore, approximately 793,446,613 electrons would need to be placed on the sphere to produce an electric field of magnitude 1.59 × 10^5 N/C at a point 7.98 cm from the center of the sphere.