Four point charges are at the corners of a square of side a as shown in the figure below. Determine the magnitude and direction of the resultant electric force on q, with ke, q, and a left in symbolic form. (Let B = 2.5q and C = 8q. Let the +x-axis be pointing to the right.)
What is the Magnitude: ___*ke*q2/a2
Well, if we're using symbols left in the equation, I guess we could say that the magnitude of the resultant electric force on q is equal to "*ke*q2/a2" candies. Because who doesn't like candy, right?
To find the magnitude of the resultant electric force on q, we need to calculate the electric force from each of the individual charges and then add them up vectorially.
Let's label the charges as follows:
A = q (located at the top left corner)
B = 2.5q (located at the top right corner)
C = 8q (located at the bottom right corner)
D = q (located at the bottom left corner)
The electric force between two point charges is given by Coulomb's Law:
F = ke * (|q1| * |q2|) / r^2
Where:
- F is the magnitude of the electric force
- ke is the electrostatic constant (ke = 8.99 x 10^9 Nm^2/C^2)
- q1 and q2 are the magnitudes of the charges
- r is the distance between the charges
Let's calculate the electric force exerted on q by each of the other charges:
1. From charge A to charge q:
The distance between A and q is a√2 (diagonal of the square). Therefore, the electric force between A and q is:
FAq = ke * (|q| * |q|) / (a√2)^2 = ke * q^2 / (2a^2)
2. From charge B to charge q:
The distance between B and q is a (side of the square). Therefore, the electric force between B and q is:
FBq = ke * (|2.5q| * |q|) / a^2 = (2.5) * (ke * q^2) / a^2 = (2.5) * ke * q^2 / a^2
3. From charge C to charge q:
The distance between C and q is a√2 (diagonal of the square). Therefore, the electric force between C and q is:
FCq = ke * (|8q| * |q|) / (a√2)^2 = (8) * (ke * q^2) / (2a^2) = 4 * ke * q^2 / a^2
4. From charge D to charge q:
The distance between D and q is a (side of the square). Therefore, the electric force between D and q is:
FDq = ke * (|q| * |q|) / a^2 = ke * q^2 / a^2
To find the resultant electric force on q, we need to find the vector sum of these forces:
Resultant Force = FAq + FBq + FCq + FDq
Now, we can substitute the values of ke, q, and a into the equation to find the magnitude of the resultant force.
To determine the magnitude of the resultant electric force on q, we need to calculate the individual electric forces exerted by each of the four point charges and then find the vector sum of these forces.
The electric force between two point charges is given by Coulomb's law:
F = ke * (|q1| * |q2|) / r^2
Where:
- F is the magnitude of the electric force
- ke is the Coulomb's constant (approximated as 8.99 * 10^9 N·m^2/C^2)
- q1 and q2 are the magnitudes of the two point charges
- r is the distance between the charges
In this case, we have four point charges arranged around q in a square. Let's label them as A, B, C, and D, with q at the center of the square.
- Point charge A has a magnitude of q, so the force exerted by A on q is given by F_A = ke * (|q| * |q|) / a^2
- Point charge B has a magnitude of 2.5q, so the force exerted by B on q is given by F_B = ke * (|2.5q| * |q|) / a^2
- Point charge C has a magnitude of 8q, so the force exerted by C on q is given by F_C = ke * (|8q| * |q|) / a^2
- Point charge D has a magnitude of q, so the force exerted by D on q is given by F_D = ke * (|q| * |q|) / a^2
Now we can find the resultant electric force by finding the vector sum of these forces:
Resultant force = F_A + F_B + F_C + F_D
Substituting the expressions for each force:
Resultant force = ke * (|q| * |q|) / a^2 + ke * (|2.5q| * |q|) / a^2 + ke * (|8q| * |q|) / a^2 + ke * (|q| * |q|) / a^2
Simplifying:
Resultant force = ke * (15.5q^2) / a^2
Therefore, the magnitude of the resultant electric force on q is:
Magnitude = ke * (15.5q^2) / a^2
Please note that the direction of the resultant electric force also needs to be determined. To do this, we need to consider the direction of each individual force and apply vector addition. If you provide more information about the direction of each charge, I can help you determine the direction of the resultant force.