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AnnZ [28]
2 years ago
11

Find the direction and magnitude of the net force exerted on the point charge q3 in the figure. Let q= +2.4 μC and d= 33cm.

Physics
2 answers:
kobusy [5.1K]2 years ago
8 0

With the use of electric force formula, the direction and magnitude of the net force exerted on the point charge q3 are 9.9 x 10^{11} N and 66 degrees

ELECTRIC FORCE (F)

F = \frac{KQq}{d^{2} }

Where K = 9 x 10^{9} Nm^{2}/C^{2}

The distance between q_{1} and q_{3} can be calculated by using Pythagoras theorem.

d = \sqrt{33^{2} + 33^{2}  }

d = 46.7 cm = 0.467 m

For force F_{1}, substitute all the parameters into the formula above

F_{1} = (9 x 10^{9} x 3 x 1)/0.467^{2}

F_{1} = 2.7 x 10^{10}/0.218

F_{1} = 1.24 x 10^{11} N

For force F_{4}, substitute all the parameters into the formula above

F_{4} = (9 x 10^{9} x 3 x 4)/0.33^{2}

F_{4} = 1.08 x 10^{11}/0.1089

F_{4} = 9.92 x 10^{11} N

For force F_{2}, substitute all the parameters into the formula above

F_{2} = (9 x 10^{9} x 3 x 2)/0.33^{2}

F_{2} = 5.4 x 10^{10}/0.1089

F_{2} = 4.96 x 10^{11} N

Summation of forces on Y component will be

F_{y} = F_{4} - F_{1} Sin 45

F_{y} = 9.92 x 10^{11} - 1.24 x 10^{11} Sin 45

F_{y} = 9.04 x 10^{11} N

Summation of forces on X component will be

F_{x} = F_{2} - F_{1} Cos 45

F_{x} = 4.96 x 10^{11} - 1.24 x 10^{11} Sin 45

F_{x} = 4.08 x 10^{11} N

Net Force = \sqrt{F_{x} ^{2} + F_{y} ^{2}  } }

Net force = \sqrt{(4.08*10^{11}) ^{2} + (9.04*10^{11}) ^{2}  }

Net force = 9.9 x 10^{11} N

The direction will be

Tan ∅ = F_{y}/F_{x}

Tan ∅ = 9.04 x 10^{11} / 4.08 x 10^{11}

Tan ∅ = 2.216

∅ = Tan^{-1}(2.216)

∅ = 65.7 degrees

Therefore, the direction and magnitude of the net force exerted on the point charge q3 are 9.9 x 10^{11} N and 66 degrees approximately.

Learn more about electric Force here: brainly.com/question/4053816

Colt1911 [192]2 years ago
5 0

The addition of vectors and Coulomb's law allows us to find the result for the force on the charge q₃ is:

  • The modulus is F = 5.71 N
  • The direction is: tes = -65º

Given parameters,

  • The value of the charges, q₁= +q, q₂= - 2 q, q₃= -3 q, q₄= -4q
  • The value of the charge q=+ 2.4 μC = 2.4 10⁻⁶ C
  • The side of the square is d = 33 cm = 0.33 m

To find.

  • The force on q3

<h3>Electric Force. </h3>

The electric force is given by Coulomb's law, which states that the force is proportional to the charges and inversely proportional to the square of the distance.

               F =k \frac{q_1q_2}{r_{12}^2}  

where F is the force, k Coulomb's constant, q the charges and r the distance between the charges.

<h3>Sum Vectors. </h3>

Force is a vector magnitude, so vector algebra must be used for vector addition, an easy and efficient way is to use an analytical method for addition:

  • We decompose each vector
  • We make the sum of the components
  • We construct the resulting vector.

In the attachment we have a diagram of the forces, the sum on each axis is:

x axis

          Fₓ= F₂- F₁ₓ

y axis

          F_y = -F_4 + F_{1y}  

Let's use trigonometry to find the component of force.

        cos 45 = \frac{F_1_x}{F_1}  

        cis 45 = \frac{F_1_y}{F_1}  

        F₁ₓ= F₁ cos 45

        F_1_y = F₁ sin 45

Let's look for the distance between the charges, for charges 2 and 3 charges 4 and 3 the distance of is equal to the side of the square

          r=r₂₃ = r₄₃ = d

The distance between charges 1 and 3 is the diagonal of the square that we can find with the Pythagorean theorem.

       R₁₃² = d² + d²

        R₁₃² = 2d²

We substitute in Coulomb's law.

        F₂₃ = k \frac{q_2q_3}{d^2}

        F₄₃ = k \frac{q_4q_3}{d^2}

         F₁₃ = k \frac{q_1q_3}{2d^2 }  

Let us substitute in the components of the force on each axis.

x axis

         Fₓ= F₂- F₁ₓ

         F_x = k \frac{q_2q_3}{d^2} - k \frac{q_1q_3}{2d^2} cos 45  \\F_x = k \frac{q_3}{d^2} ( q_2 - \frac{q_1 cos 45}{2})

y axis

          F_y = - F_4 + F_1_y  \\F_y = k \frac{q_4q_3}{d^2} - k \frac{q_1q_3}{2d^2} sin 45 \\F_y = k \frac{q_3}{d^2} ( -q_4 + \frac{q_1sin 45}{2})

We substitute in value of each charge.

x axis

           F_x = k \frac{3q}{d}  \ q( 2 - \frac{1 \ cos 45}{2} ) \\F_x = 3k (\frac{q}{d})^2  1.6464

y axis

         F_y = k \frac{3q}{d^2} ( -4 + \frac{1 \ sin45}{2} ) \\F_y = 3k  (\frac{q}{d})^2 ( -3.64645)

The resulting vector is

          F = F_x \hat i + f_y \hat j  

          F = 3k(\frac{q}{d} )^2 ( 1.6464 \hat i - 3.64645 \hat j )3k q² /d² ( 1.6464 i^ - 3.64645 j^ )

We calculate.

        F = 3 9 10⁹ ( 2.4 10-6/0.33)² ( 1.6464 i^ - 3.64645 j^ )

         F = 1.428( 1.6464 i⁻ 3.64645 j^

We build the resulting vector, for the module we use the Pythagorean theorem.

          F = \sqrt{F_x^2 + F_y^2}\\F = 1.428 \ \sqrt{1.6464^2+ 3.64645^2}

          F = 5.71N

Let's use trigonometry for the angle.

        tan \theta = \frac{F_y}{F_x}  

        \theta = tan^{-1}  (\frac{-3.64645}{1.6464} )  

        θ = -65º

This angle is half clockwise from the positive side of the x-axis.

in conclusion using vector addition and Coulomb's law we can find the result for the force on the charge q3 is:

  • The direction is: tes = -65º
  • The modulus is F = 5.71 N

Learn more about the electric force here: brainly.com/question/26153267

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