Two particles are fixed along a straight line, separated by a distance 6.0 cm. The left one has a charge +2.0 nC and the right one has a charge of +6.0 nC. You must try to locate a point where the electric field is zero.

(a) What is the location of the point?
(b) How far from the left charge is the point?
(c) Suppose that the left charge is -2 nC. Now where would you find a point where the electric field is zero??
(d) How far is the point measured from the left charge?

kq₁/x²=kq₂/(6-x)²

2/x²=6/(6-x)²
36-12x+x²=3x²
2x²+12x-36=0
x²+6x-18=0
x = - 3±sqrt(9+18) =
= - 3 ± 5.2
If q₁ >0 and q₂>0 x= 2.2 cm (to the right from the left charge (between the charges))
If q₁ <0 and q₂>0 x=-8.2 cm (to the left from the left (negative) charge)

To find the location of a point where the electric field is zero in this scenario, we can use the concept of electric field due to point charges. Electric field is a vector quantity that represents the force per unit charge experienced by a positive test charge placed in the field.

(a) To find the location of the point where the electric field is zero, we need to consider the electric fields produced by each charge separately and determine where their effects cancel each other out. The formula to calculate the electric field due to a point charge is given by:

Electric Field (E) = k * (q/r^2)

Where:
- E is the electric field
- k is Coulomb's constant (9.0 x 10^9 N m²/C²)
- q is the charge of the point charge
- r is the distance from the point charge to the location where the electric field is being calculated.

For simplicity, let's label the left charge as Charge 1 (q1 = +2.0 nC) and the right charge as Charge 2 (q2 = +6.0 nC). We want the electric field to be zero, so the electric field due to Charge 1 (-E1) must be equal in magnitude but opposite in direction to the electric field due to Charge 2 (E2).

Therefore, we have the following equation:

-E1 = E2

Substituting the values into the formula, we have:

- k * (q1/r^2) = k * (q2/(6.0 cm - r)^2)

Simplifying the equation, we can cancel out the Coulomb's constant (k) and the distance from Charge 2 (6.0 cm - r)^2. This leaves us with:

q1/r^2 = q2/(6.0 cm - r)^2

Now, we can substitute the values of the charges:

(2.0 nC)/r^2 = (6.0 nC)/(6.0 cm - r)^2 [1]

(b) To find the distance from the left charge to the point where the electric field is zero, we solve equation [1] for r.

To simplify the calculation, we can cross-multiply:

(2.0 nC)(6.0 cm - r)^2 = (6.0 nC)(r^2)

Expanding the equation:

12.0 cm^2 - 12.0 cmr + r^2 = 6.0 cmr^2

Rearranging terms:

6.0 cmr^2 + 12.0 cmr - 12.0 cm^2 + r^2 = 0

Combining like terms:

6.0 cmr^2 + (12.0 cm - 12.0 cm)r + r^2 - 12.0 cm^2 = 0

Simplifying:

6.0 cmr^2 + 12.0 cmr + r^2 - 12.0 cm^2 = 0

Now we can factorize the quadratic equation:

(2r - 6.0 cm)(3r - 2.0 cm) = 0

Setting each factor equal to zero and solving for r:

2r - 6.0 cm = 0 or 3r - 2.0 cm = 0

2r = 6.0 cm or 3r = 2.0 cm

r = 3.0 cm or r = 0.67 cm

Since the distance cannot be negative, the possible values for r are 3.0 cm or 0.67 cm.

(c) If the left charge is changed to -2 nC, we can repeat the same steps as before to find the new distance where the electric field is zero.

Now, equation [1] becomes:

(-2.0 nC)/r^2 = (6.0 nC)/(6.0 cm - r)^2

Following the same process, we find the new value for r.

(d) Once we have the values for r, we can use the distance from the left charge (q1) to calculate the distance from the point where the electric field is zero.

If r = 3.0 cm, the point is located 3.0 cm away from the left charge.
If r = 0.67 cm, the point is located 0.67 cm away from the left charge.