a square with side length 2.0 cm has a charge of -1.0x10^-6C at every corner. What is the magnitude and direction of the electric force on each charge?
What is the force on a fifth charge placed in the centre of this square?
Does the sign of the fifth charge affect the magnitude or direction of force on it?
Make the drawing:
charge #1 – at the left bottom corner,
charge #2 – at the left upper top cirner,
charge #3 – at the right top corner,
charge #4 – at the right bottom corner.
Forces actting on the Charge #4 are:
from #1 towards #4 F1=k•q1•q2/a²=k•q²/a² ,
from #3 towards #4 F3 = k•q1•q2/a²=k•q²/a² ,
from #2 towards #4 F2 = k•q1•q2/(a√2)²=k•q²/2•a² ,
F13=sqrt(F1² +F3²)= k•√2•q²/a².
F123= F13+F2 =k•√2•q²/a² + k•q²/2•a² =
= k•q²/ a•(√2+0.5) =9•10^9•10^-12•1.91/0.02 =0.86 N.
The equal net forces act on each charge.
The electric field strength is zero in the center of this square, therefore, the force acts on each charge in the center is zero as well.
To calculate the magnitude and direction of the electric force on each charge at the corners of the square, we can use Coulomb's Law.
Coulomb's Law states that the magnitude of the electric force between two charges is given by:
F = k * (|q1| * |q2|) / r^2
Where:
F = Electric force
k = Coulomb's constant (8.99 x 10^9 N m^2/C^2)
|q1| and |q2| = Magnitude of the charges
r = Distance between the charges
Given:
Charge at each corner: -1.0x10^-6 C
Side length of the square: 2.0 cm = 0.02 m
Calculating the electric force on each charge:
Since all charges at the corners of the square are the same, we can calculate the force between any two charges and then multiply it by 4 to get the total force on each charge.
Let's consider one corner charge:
|q1| = |q2| = 1.0x10^-6 C (converting -1.0x10^-6 C to positive magnitude)
r = 0.02 m (distance between two charges)
Using Coulomb's Law:
F = (8.99 x 10^9 N m^2/C^2) * (1.0x10^-6 C)^2 / (0.02 m)^2
= 8989 N
Therefore, the magnitude of the electric force on each charge at the corners of the square is 8989 N.
To determine the direction of the electric force on each charge, we can use the principle that like charges repel each other. Since all corner charges have the same magnitude, the electric force will be repulsive between all pairs of charges. The direction of the force on each charge will be away from the other charges.
Now, let's calculate the force on a fifth charge placed in the center of the square:
The magnitude of the force on the fifth charge can be calculated using Coulomb's Law, considering one of the corner charges and the fifth charge:
|q1| = |q2| = 1.0x10^-6 C (charge at the corner)
|q3| = Magnitude of the fifth charge (unknown)
r = 0.02 m (distance between the corner charge and the fifth charge)
Using Coulomb's Law:
F = (8.99 x 10^9 N m^2/C^2) * (1.0x10^-6 C) * |q3| / (0.02 m)^2
The sign of the fifth charge will affect both the magnitude and direction of the force on it. If the fifth charge is positive, it will experience a repulsive force from the negatively charged corner charges. If the fifth charge is negative, it will experience an attractive force towards the negatively charged corner charges.
In summary:
- The magnitude of the electric force on each charge at the corners of the square is 8989 N, and the direction of the force is away from the other charges.
- The force on a fifth charge placed in the center of the square can be calculated using Coulomb's Law, and its magnitude and direction will depend on the sign of the fifth charge.
To calculate the magnitude and direction of the electric force on each charge at the corners of the square, we can use Coulomb's Law. Coulomb's Law states that the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
First, let's calculate the magnitude of the electric force on each corner charge.
Given:
- Charge of each corner = -1.0x10^-6 C
- Side length of the square = 2.0 cm
Step 1: Convert the side length from centimeters to meters.
1 meter = 100 centimeters
So, 2.0 cm = 2.0 / 100 = 0.02 meters
Step 2: Calculate the distance between the corner charges.
The distance between opposite corners of the square is equal to the length of its diagonal. In this case, the diagonal is equal to the side length times the square root of 2.
Diagonal of the square = 0.02 * √2 meters
Step 3: Calculate the magnitude of the electric force.
Using Coulomb's Law formula:
F = k * (|q1| * |q2|) / r^2
Where:
F is the magnitude of the electric force,
k is the Coulomb's 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 calculate the magnitude of the electric force on each corner charge at one of the corners of the square (the other corners will have the same magnitude of force):
F = (8.99 x 10^9 N m^2/C^2) * (|-1.0x10^-6 C| * |-1.0x10^-6 C|) / (0.02 * √2 m)^2
Calculating this, we get:
F ≈ 2.03 x 10^-2 N (rounded to two decimal places)
Since all the corner charges are of the same magnitude (-1.0x10^-6 C) and at the same distance from each other, the magnitude of the electric force on each corner charge will also be approximately 2.03 x 10^-2 N, pointing inwards towards the center of the square.
Now, let's consider the fifth charge placed in the center of the square. The magnitude and direction of the force on this charge will be determined by the other corner charges.
Since the corner charges are negatively charged and the fifth charge is placed in their midst, it will experience an attractive force in the direction towards each corner charge. The magnitude of the force on the fifth charge will be four times the magnitude of the force on each corner charge, since there are four corner charges.
Therefore, the magnitude of the force on the fifth charge will be approximately 4 * 2.03 x 10^-2 N, which simplifies to approximately 8.12 x 10^-2 N (rounded to two decimal places).
The sign of the fifth charge will affect the direction of the force on it. If the fifth charge is negatively charged, it will experience an attractive force towards each corner charge. If the fifth charge is positively charged, it will experience a repulsive force away from each corner charge. However, the magnitude of the force on the fifth charge will remain the same regardless of its sign.