Two point charges are 2.00 m apart as shown on the right. The charge of the first object is -1.50 C, while the second object has a charge of +6.85 C. At which point will the electric field vector be zero?

in simply steps

To find the point at which the electric field vector is zero, we can use the principle that the electric field due to two point charges is given by the vector sum of the electric fields due to each individual charge.

First, let's denote the position of the first object with charge -1.50 C as Q1, and the position of the second object with charge +6.85 C as Q2.

Next, let's consider a point P at which we want the electric field vector to be zero.

The electric field due to Q1 at point P is given by the formula:

E1 = (k * |Q1|) / (r1^2)

Where k is the electrostatic constant (approximately 9 x 10^9 N m^2/C^2), |Q1| is the magnitude of the charge of Q1 (1.50 C) and r1 is the distance from Q1 to point P.

Similarly, the electric field due to Q2 at point P is given by:

E2 = (k * |Q2|) / (r2^2)

Where |Q2| is the magnitude of the charge of Q2 (6.85 C) and r2 is the distance from Q2 to point P.

Since we want the electric field vector to be zero at point P, the vector sum of E1 and E2 should be zero:

E1 + E2 = 0

Substituting the expressions for E1 and E2, we have:

(k * |Q1|) / (r1^2) + (k * |Q2|) / (r2^2) = 0

Simplifying this equation, we get:

(|Q1| / r1^2) + (|Q2| / r2^2) = 0

Now, we can rearrange this equation to solve for r2^2:

(|Q2| / r2^2) = - (|Q1| / r1^2)

r2^2 / |Q2| = -r1^2 / |Q1|

Taking the square root of both sides:

r2 / sqrt(|Q2|) = -r1 / sqrt(|Q1|)

To find the point at which the electric field vector is zero, we need to find the ratio of the distances from the charges to that point:

r2 / r1 = - sqrt(|Q2|) / sqrt(|Q1|)

Plugging in the given values:

r2 / 2.00m = - sqrt(6.85 C) / sqrt(1.50 C)

Simplifying this equation gives us the ratio of distances:

r2 / 2.00m = -1.98

Finally, we can solve for r2 by multiplying both sides by 2.00m:

r2 = -3.96m

Therefore, the electric field vector will be zero at a point that is -3.96m away from the second charge, Q2.

To find the point at which the electric field vector is zero, we need to set up an equation based on the electric field due to each point charge and solve for the position.

Step 1: Determine the direction of the electric field due to each charge:
- The electric field due to a negative charge points towards the charge.
- The electric field due to a positive charge points away from the charge.

Step 2: Calculate the electric field due to each charge at a general location (x) from the first charge:
- For the first charge, the electric field (E1) can be calculated using Coulomb's law:
E1 = k*q1/r^2
where k is the electrostatic constant (9.0 x 10^9 N·m^2/C^2), q1 is the charge of the first object (-1.50 C), and r is the distance between the charges.

- For the second charge, the electric field (E2) can also be calculated using Coulomb's law:
E2 = k*q2/r^2
where q2 is the charge of the second object (+6.85 C), and r is the distance between the charges.

Step 3: Write an equation for the total electric field at a general location (x):
- The total electric field, E_total, is the vector sum of the electric fields due to each charge:
E_total = E1 + E2

Step 4: Set up the equation for E_total = 0 and solve for x:
0 = E1 + E2

Step 5: Substitute the expressions for E1 and E2 into the equation:
0 = k*q1/x^2 + k*q2/(2.00 - x)^2

Step 6: Solve the equation for x:
Rearranging the equation, we get:
k*q1/x^2 = -k*q2/(2.00 - x)^2

Cross-multiplying and simplifying, we have:
k*q1*(2.00 - x)^2 = -k*q2*x^2

Expanding and rearranging the equation further:
(2.00 - x)^2 = -q2*q1*x^2/q1

Taking the square root of both sides, we get:
2.00 - x = ±sqrt(-q2*q1)/x (since x cannot be zero)

Simplifying the right side:
2.00 - x = ±sqrt(q2*q1)/x

Now, we can solve for x by rearranging the equation:
2.00 = x ± sqrt(q2*q1)/x

We can further simplify this equation by multiplying both sides by x:
2.00x = x^2 ± sqrt(q2*q1)

Rearranging and combining like terms:
x^2 - 2.00x ± sqrt(q2*q1) = 0

Step 7: Solve the quadratic equation for x:
Using the quadratic formula, x = (-b ± sqrt(b^2 - 4ac)) / 2a, we can identify a, b, and c as follows:
a = 1, b = -2.00, c = ± sqrt(q2*q1)

Substituting these values into the quadratic formula and evaluating for x, we can find the two possible solutions.

Step 8: Check the validity of the solutions:
Check if the calculated values of x are within the range of 0 to 2.00. If they are, they are valid solutions representing points between the two charges.

Therefore, by following these steps, you can find the point at which the electric field vector is zero.

To find the point at which the electric field vector is zero, we can use the principle that the electric field at a given point is the vector sum of the electric fields due to each charge. In this case, since we only have two charges, we can equate the electric fields due to each charge and solve for the position.

Let's denote the position of the first charge as q1 with a charge of -1.50 C, and the position of the second charge as q2 with a charge of +6.85 C.

1. Define a coordinate system: Assign one of the charges, say q1, as the origin. Let's assume that the positive x-direction is to the right.

2. Use Coulomb's Law: The electric field (E) due to a point charge can be calculated using Coulomb's Law:

E = k * (q / r^2)

Where
E is the electric field,
k is the electrostatic constant (9.0 x 10^9 Nm^2/C^2),
q is the charge, and
r is the distance between the point and the charge.

3. Determine the electric fields due to each charge: Calculate the electric fields due to each charge at a general point (x, 0) on the x-axis.

E1 = k * (q1 / (x^2))
E2 = k * (q2 / (2.00 - x)^2)

4. Set up the equation for the sum of electric fields: Since the electric field is a vector quantity, the sum of the electric fields from each charge must be zero at the desired point. Therefore, we equate the magnitudes of the two electric fields:

|E1| = |E2|

k * (q1 / (x^2)) = k * (q2 / (2.00 - x)^2)

Simplifying this equation, we get:

q1 / (x^2) = q2 / (2.00 - x)^2

5. Solve the equation: Cross multiply and simplify the equation to solve for x.

q1 * (2.00 - x)^2 = q2 * x^2

Solve this equation for x. You can expand the squared terms, collect like terms, and then bring everything to one side of the equation. Then, you can use methods like factoring, quadratic formula, or graphical analysis to find the value(s) of x.

6. Check the solution: Once you find the value of x, substitute it back into the equation for the electric fields to confirm that the sum of the electric fields is indeed zero.

E1 = k * (q1 / (x^2))
E2 = k * (q2 / (2.00 - x)^2)

Calculate the magnitudes of the electric fields using the given values of q1, q2, and x. If the magnitudes of both electric fields are equal and opposite, the electric field vector will be zero at that point.

By following these steps, you can determine the point at which the electric field vector is zero for the given scenario.