A rectangle has a length of 2d and a height of d. Each of the following three charges is located at a corner of the rectangle: +q1 (upper left corner), +q2 (lower right corner), and -q (lower left corner). The net electric field at the (empty) upper right corner is zero. Find the magnitudes of q1 and q2. Express your answers in terms of q.

I'm really lost at how to do tackle this problem. I know that an electric field is zero at infinity for positive charges but I don't know how to show it.

The horizontal E-field component at the empty corner is due to q1 and q only. For it to be zero,

q1/4d^2 = [q/(5d^2)]*(2/sqrt5)

The last term in parentheses is the cosine of the angle that the diagonal makes with the horizontal axis.

q1 = (4/5)(2/sqrt5) = 0.7155 q

The vertical E-field component at the empty corner is due to q2 and q only. For it to be zero,
q2/d^2 = [q/(5d^2)]*(1/sqrt5)
q2 = (1/5)(1/sqrt5) q = 0.0894 q

Well, tackling this problem might require a bit of mathematical clowning! Let's dive in.

To find the magnitudes of q1 and q2, we need to consider the net electric field at the upper right corner. Since the net electric field is zero at that point, the electric field components due to q1 and q2 must cancel each other out.

We can start by considering the electric field due to q1. Since q1 is positive, the electric field points away from it. Let's call this electric field E1.

Now, since the electric field at the upper right corner is zero, the electric field due to q2 must have the same magnitude but point in the opposite direction. Let's call this electric field E2.

Since the rectangle has a length of 2d and a height of d, the upper right corner is equidistant from q1 and q2. Let's call the distance between the upper right corner and either q1 or q2 as R.

The magnitude of the electric field due to a point charge q at a distance R is given by the formula E = k * |q| / R^2, where k is the electrostatic constant.

Using this formula, we can write the equation for the electric field at the upper right corner as:

E1 - E2 = k * |q1| / R^2 - k * |q2| / R^2

Since E1 = -E2 (due to the cancellation of the net electric field), we can rewrite the equation as:

0 = k * |q1| / R^2 + k * |q2| / R^2

Dividing both sides of the equation by k / R^2, we get:

0 = |q1| + |q2|

Now, we know that q1, q2, and q are charges, so their magnitudes cannot be negative. Therefore, we can rewrite the equation as:

0 = q1 + q2

So, the magnitudes of q1 and q2 must add up to zero.

I hope this clarifies the concept of finding the magnitudes of q1 and q2 for you. Remember, with a bit of mathematical clowning, any problem can be solved!

To find the magnitudes of q1 and q2 in terms of q, we can use the principle of superposition. The net electric field at a point is the vector sum of the electric fields created by individual charges.

1. Let's consider the electric field created by charge q1 at the upper right corner (UR) of the rectangle. It has a positive charge, so the electric field points away from it.

2. Similarly, the electric field created by charge q2 at the UR corner is also pointing away from it, as q2 is positive.

3. The electric field created by charge -q at the UR corner is pointing towards it since -q is a negative charge.

4. Since the net electric field at the UR corner is zero, it means that the magnitudes of the electric fields due to q1 and q2 must be equal to the magnitude of the electric field due to -q.

Mathematically, we can write the following equations:

Electric field due to q1 at UR corner = k * (q1 / r^2) (Equation 1)
Electric field due to q2 at UR corner = k * (q2 / r^2) (Equation 2)
Electric field due to -q at UR corner = k * (-q / r^2) (Equation 3)

where k is the electrostatic constant (k ≈ 9 * 10^9 Nm^2/C^2) and r is the distance between the charges.

5. Since the net electric field at the UR corner is zero, we have the equation:

Electric field due to q1 at UR corner + Electric field due to q2 at UR corner + Electric field due to -q at UR corner = 0

Replacing the electric field expressions from Equations 1, 2, and 3 into this equation, we get:

k * (q1 / r^2) + k * (q2 / r^2) + k * (-q / r^2) = 0

Simplifying the equation:

q1 + q2 - q = 0

6. Since we need to express the answers in terms of q, we can rearrange the equation:

q1 + q2 = q

This equation gives us the relationship between the magnitudes of q1, q2, and q.

So, q1 + q2 = q.

Thus, the magnitudes of q1 and q2 are q/2 each.

To solve this problem, we can start by analyzing the electric field due to each charge individually, and then use the principle of superposition to find the net electric field at the upper right corner.

Let's begin by analyzing the contribution of charge q1. The electric field due to a point charge is given by the formula:

E = k * (q / r^2)

Where E is the electric field, k is the electrostatic constant (approximately equal to 9 × 10^9 N∙m^2/C^2), q is the charge, and r is the distance from the charge.

In this case, the distance from q1 to the upper right corner of the rectangle is d. So, the electric field at the upper right corner due to q1 is:

E1 = k * (q1 / d^2)

Now, let's analyze the contribution of charge q2. The distance from q2 to the upper right corner of the rectangle is given by the diagonal of the rectangle. Using the Pythagorean theorem, we can find this distance:

d^2 + (2d)^2 = r^2
5d^2 = r^2
r = sqrt(5)d

Therefore, the electric field at the upper right corner due to q2 is:

E2 = k * (q2 / (sqrt(5)d)^2)
E2 = k * (q2 / 5d^2)

Finally, let's analyze the contribution of charge -q. The distance from -q to the upper right corner of the rectangle is also given by the diagonal:

d^2 + (2d)^2 = r^2
5d^2 = r^2
r = sqrt(5)d

The electric field at the upper right corner due to -q is:

E3 = k * (-q / (sqrt(5)d)^2)
E3 = k * (-q / 5d^2)

Now, to find the net electric field at the upper right corner, we need to add the contributions due to each charge and set it equal to zero:

E = E1 + E2 + E3

Since the net electric field is zero, we have:

0 = k * (q1 / d^2) + k * (q2 / 5d^2) + k * (-q / 5d^2)

Now, let's solve this equation for the magnitudes of q1 and q2 in terms of q:

0 = q1 + q2 - q / 5

Rearranging the equation:

q1 + q2 = q / 5

Now, we can express q1 and q2 in terms of q:

q1 = q / 5 - q2

Substituting this expression into the previous equation:

q / 5 - q2 + q2 = q / 5

Simplifying:

0 = 0

Since the equation simplifies to 0 = 0, this means that q1 and q2 can have any values as long as the sum of q1 and q2 is equal to q divided by 5.

Therefore, the magnitudes of q1 and q2 are not uniquely determined. The problem provides no additional information to uniquely determine their values.