Seven charges are
placed on the corners of a regular hexagon and its center. They all have a charge magnitude of Q=
4.15 µC. If the five charges (Solid Black) are positively charged and two negatively charged (White).
Note: The length of one side of the hexagon is 5.00x101
cm.
a) Find the net acting, Fnet on charge q7.
b) If charge q7 is removed from the center,
b.i. Calculate the net electric field, Enet at that center.
b.ii. Calculate the electric potential Vtotal at that center
a) To find the net acting force on charge q7, we need to calculate the vector sum of the individual forces exerted by the other charges.
We can start by calculating the magnitude and direction of the force between each pair of charges using Coulomb's Law:
F = (k * |q1| * |q2|) / r^2
where k is the electrostatic constant (8.99 x 10^9 N m^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
Let's label the charges on the corners of the hexagon (in a clockwise direction) as q1, q2, q3, q4, q5, and q6, and the charge at the center as q7.
The magnitude of each charge is |q1| = |q2| = |q3| = |q4| = |q5| = |q6| = 4.15 µC = 4.15 x 10^-6 C.
The magnitude of charge q7 is also |q7| = 4.15 x 10^-6 C.
The distance between each corner charge and the center charge is the length of one side of the hexagon, which is 5.00 x 10^-1 m.
Now we can calculate the force between each pair of charges:
F1-7 = (k * |q1| * |q7|) / r^2
F2-7 = (k * |q2| * |q7|) / r^2
F3-7 = (k * |q3| * |q7|) / r^2
F4-7 = (k * |q4| * |q7|) / r^2
F5-7 = (k * |q5| * |q7|) / r^2
F6-7 = (k * |q6| * |q7|) / r^2
The net force on charge q7 is the vector sum of these individual forces:
Fnet = F1-7 + F2-7 + F3-7 + F4-7 + F5-7 + F6-7
To calculate the force vector, we need to determine the direction of each individual force.
Since all the corner charges are positively charged and as opposite charges attract, the forces between charges q1-q7, q2-q7, q3-q7, q4-q7, q5-q7, and q6-q7 are all inward, towards the center charge.
Therefore, the net acting force on charge q7 is the vector sum of these inward forces.
b) If charge q7 is removed from the center, the net electric field at that center can be calculated using the principle of superposition.
The net electric field at a point is the vector sum of the electric fields produced by each individual charge.
The electric field produced by a single charge is given by:
E = (k * |q|) / r^2
where k is the electrostatic constant, |q| is the magnitude of the charge, and r is the distance between the charge and the point where the electric field is being measured.
Let's consider the electric field produced by each individual charge at the center of the hexagon.
E1 = (k * |q1|) / r^2
E2 = (k * |q2|) / r^2
E3 = (k * |q3|) / r^2
E4 = (k * |q4|) / r^2
E5 = (k * |q5|) / r^2
E6 = (k * |q6|) / r^2
The net electric field Enet at the center is the vector sum of these individual electric fields:
Enet = E1 + E2 + E3 + E4 + E5 + E6
To calculate the electric potential Vtotal at the center, we can use the principle that the electric potential at a point is the sum of the electric potentials due to each individual charge.
The electric potential due to a single charge is given by:
V = (k * |q|) / r
where k is the electrostatic constant, |q| is the magnitude of the charge, and r is the distance between the charge and the point where the electric potential is being measured.
The total electric potential at the center is the sum of the individual electric potentials:
Vtotal = V1 + V2 + V3 + V4 + V5 + V6
a) To find the net force acting on charge q7, we need to calculate the individual forces between q7 and each of the other charges.
Since each of the five solid black charges has a positive charge, they will repel the charge q7. The two negatively charged white charges will attract the charge q7.
The magnitude of the electric force between two charges can be calculated using Coulomb's law:
F = k * (|q1| * |q2|) / r²
where F is the electric force, k is the electrostatic constant (8.99 x 10^9 N m²/C²), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
Since we have a regular hexagon, the distance between q7 and each of the other charges is the length of one side of the hexagon (5.00 x 10¹ cm).
Using Coulomb's law, the force between q7 and each of the five positive charges is:
F1 = k * (|q7| * |q1|) / r²
F2 = k * (|q7| * |q2|) / r²
F3 = k * (|q7| * |q3|) / r²
F4 = k * (|q7| * |q4|) / r²
F5 = k * (|q7| * |q5|) / r²
Since the two negatively charged white charges attract q7, the forces will have opposite directions:
F6 = -k * (|q7| * |q6|) / r²
F7 = -k * (|q7| * |q7|) / r²
To find the net force, we add up all seven forces:
Fnet = F1 + F2 + F3 + F4 + F5 + F6 + F7
Substituting the given values (Q = 4.15 µC, r = 5.00 x 10¹ cm, k = 8.99 x 10^9 N m²/C²) and calculating the forces will give us the net force on charge q7.
b) i) If charge q7 is removed from the center, we can calculate the net electric field at that center.
The net electric field at a point can be calculated by adding up the individual electric fields due to each of the charges.
The magnitude of the electric field due to a point charge can be calculated using the equation:
E = k * |q| / r²
where E is the electric field, k is the electrostatic constant (8.99 x 10^9 N m²/C²), |q| is the magnitude of the charge, and r is the distance from the charge.
Since the distance between the center and each of the charges is the length of one side of the hexagon (5.00 x 10¹ cm), we can calculate the electric fields due to each charge:
E1 = k * |q1| / r²
E2 = k * |q2| / r²
E3 = k * |q3| / r²
E4 = k * |q4| / r²
E5 = k * |q5| / r²
E6 = k * |q6| / r²
E7 = k * |q7| / r²
To find the net electric field at the center, we add up all seven electric fields:
Enet = E1 + E2 + E3 + E4 + E5 + E6 + E7
Substituting the given values (Q = 4.15 µC, r = 5.00 x 10¹ cm, k = 8.99 x 10^9 N m²/C²) and calculating the electric fields will give us the net electric field at the center.
b) ii) To calculate the electric potential (Vtotal) at the center, we need to sum the electric potentials due to each of the charges.
The electric potential (V) due to a point charge can be calculated using the equation:
V = k * |q| / r
where V is the electric potential, k is the electrostatic constant (8.99 x 10^9 N m²/C²), |q| is the magnitude of the charge, and r is the distance from the charge.
Since the distance between the center and each of the charges is the length of one side of the hexagon (5.00 x 10¹ cm), we can calculate the electric potentials due to each charge:
V1 = k * |q1| / r
V2 = k * |q2| / r
V3 = k * |q3| / r
V4 = k * |q4| / r
V5 = k * |q5| / r
V6 = k * |q6| / r
V7 = k * |q7| / r
To find the total electric potential at the center, we add up all seven electric potentials:
Vtotal = V1 + V2 + V3 + V4 + V5 + V6 + V7
Substituting the given values (Q = 4.15 µC, r = 5.00 x 10¹ cm, k = 8.99 x 10^9 N m²/C²) and calculating the electric potentials will give us the total electric potential at the center.