Question

Two point charges (q1 = -4.8μC and q2 = 6.7 μC) are fixed along the x-axis, separated by a distance d = 7.6 cm. Point P is located at (x,y) = (d,d).

1)

What is Ex(P), the value of the x-component of the electric field produced by q1 and q2 at point P?N/C

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2)

What is Ey(P), the value of the y-component of the electric field produced by q1 and q2 at point P?N/C

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3)

h2_pointC

A third point charge q3 = 3.1 μC is now positioned along the y-axis at a distance d = 7.6 cm from q1 as shown. What is Ex(P), the x-component of the field produced by all 3 charges at point P?

N/C

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4)

Suppose all charges are now doubled (i.e., q1 = -9.6 μC, q2 = 13.4 μC, q3 = 6.2 μC), how will the electric field at P change?

Its magnitude will increase by less than a factor of two and its direction will remain the same

Its magnitude will increase by less than a factor of two and its direction will change

Its magnitude will double and its direction will remain the same

Its magnitude will double and its direction will change

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5)

How would you change q1 (keeping q2 and q3 fixed) in order to make the electric field at point P equal to zero?

Increase its magnitude and change its sign

Decrease its magnitude and change its sign

Increase its magnitude and keep its sign the same

Decrease its magnitude and keep its sign the same

There is no change you can make to q1 that will result in the electric field at point P being equal to zero.

Answer 1

Answer:

1)  Eₓ = 8.99 10³3 (6.7 /(d-0.076)² - 4.8 / d²) ,

2)   E_{y}= 1,708 10⁴/d²      N /C

3)  Eₓ = 8.99 10³ (-0.7/d² + 6.7 / (d-0.076)²)

4) magnitud is double and maintains the same angle

5) There is no change that can be made.

Explanation:

The electric field is a vector quantity with formula

           E = k q / r²

Where k is the Coulomb constant with value 8.99 10⁹ N m² / C², q is the load and r is the distance from the load to the point of interest.

The total field is

              E = E1 + E2

             E = k q₁ / r₁² + k q₂ / r₂²

 

Since the field is a vector quantity we will look for each component

1) x axis

Let's look for the distance for simplicity, suppose the origin of the coordinate system is at point 1

          x₁ = x- x₀

          x = d - 0 = d

Point 2

          x₂ = d- 0.076

          Eₓ = E₁ₓ + E₂ₓ

          Eₓ = k q₁ / x₁² + k q₂ / x₂²

          Eₓ = 8.99 10⁹ (- 4.8 10⁻⁶/ d² + 6.7 10⁻⁶ / (d-0.076)² )

          Eₓ = 8.99 10³3 (6.7 /(d-0.076)² - 4.8 / d²)

.) Y axis

Distance

The points are on the x-axis so the y coordinate of the charges is zero

          y₁ = y-y₀

          y₁ = d-0

          y₂ = d-0

         

          $E_{y}$ = E_{1y} + E_{2y}

          E_{y} = k q₁ / y₁² + k q₂ / y₂²

          E_{y} = 8.99 10⁹ (-4.8 10⁻⁶ / d² + 6.7 10⁻⁶ / d²)

           E_{y} = 8.99 10³ (1.9 / d²)

           E_{y}= 1,708 10⁴/d²      N /C

For explicit value of the electric field we must know the coordinates of point P

3) A third load is placed at a point d = 7.6 cm, on the y-axis, we calculate the field on the x-axis

Distance

           x₃ = d - 0 = d

           Eₓ = E₁ₓ + E₂ₓ + E₃ₓ

           Eₓ = k (q₁ / x₁² + q₂ / x₂² + q₃ / x₃²)

           Eₓ = 8.99 10⁹ (- 4.8 / d² + 6.7 / (d-0.076)² + 3.1 / d²) 10⁻⁶

           Eₓ = 8.99 10³ (-0.7/d² + 6.7 / (d-0.076)²)

4) The electric field at point P is

               E₀ = RA (Eₓ² + $E_{y}$²)

        With the new charges

               Eₓ = 2 Eₓ₀

               E_{y} = 2 E_{yo}

               E = √ (4Exo + 4E_{yo})

               E = 2 E₀

As the two components increase the same amount the angle does not change

The answer:  magnitud is double and maintains the same angle

5) For the field to be zero at point P the two components must be zero

  Y Axis  

          Ey = 8.99 10⁹ (-4.8 10⁻⁶/d² + 6.7 10⁻⁶ / d²)

For this amount to be zero you must increase q₁ by keeping your sign

   X axis

         Ex = 8.99 10⁹ (- 4.8 / d² + 6.7 / (d-0.076)² + 3.1 / d²) 10⁻⁶

For this component you must also increase q1 without changing sign

But we have a problem, the magnitude of the change is different on each axis and, so there is no way to take the point / p to a zero field

The serious answer: There is no change that can be made.