Charges 2 µC, -3 µC, and 4 µC are placed in the air at the vertices of an equilateral triangle of sides 10 cm. What is the magnitude of the resultant force acting on a 4 µC charge? (Answer 15.7 N)
Given:
1st charge = q₁ = 2 µC = 2 x 10⁻⁶ C
2nd charge = q₂ = -3 µC = -3 x 10⁻⁶ C
3rd charge = q₃ = 4 µC = 4 x 10⁻⁶ C
Separation distance = r = 10 cm = 0.1 m
∴ Coulomb's Constant = K = 14ㄫε0 = 9 x 10⁹ Nm²C⁻²
Separation distance = r = 10 cm = 0.1 m
∴ Coulomb's Constant = K = 14ㄫε0 = 9 x 10⁹ Nm²C⁻²
To Find:
The magnitude of the resultant force acting on q₃ = F = ?
Where F₁ is the force of repulsions between charges q₁ and q₃ (as both are +ve charges) and F₂ is the force of attractions between charges q₂ and q₃ (as q₂ is -ve and q₃ is +ve charge)
F₁ cos 60° and F₂cos 60° are the horizontal X-components of the force F₁ and F₂ respectively. Similarly F₁ sin 60° and F₂ sin 60° are the vertical y-component of the force F₁ and F₂ respectively
Here we will first calculate F₁, the force of repulsion between charges q₁ and q₃ So using coulomb's Law formula
F₁ = 14ㄫε0q₁q₃r2
by putting values
F₁= 9 x 10⁹ Nm²C⁻² x 2x10⁻⁶Cx4x10⁻⁶C(0.1m)2
F₁= 9 x 10⁹ Nm²C⁻² x 8x10⁻¹²C0.01m2
F₁ = 72x10⁹⁻¹²Nm²0.01m2
F₁ = 72x10⁻³Nm²0.01m2
F₁ = 7200 x 10⁻³ N
F₁ = 7.2 N ---------------(1).
and
F₂, the force of attraction between charges q₂ and q₃ So using coulomb's Law formula again
F₂ = 14ㄫε0q₂q₃r2
by putting values
F₂ = 9 x 10⁹ Nm²C⁻² x -3x10⁻⁶Cx4x10⁻⁶C(0.1m)2
F₂ = 9 x 10⁹ Nm²C⁻² x -12x10⁻¹²C0.01m2
F₂ = - 108x10⁹⁻¹²Nm²0.01m2
F₂ = -108x10⁻³Nm²0.01m2
F₂ = -10800 x 10⁻³ N
F₂ = -10.80 N ---------------(2).
The -ve sign just shows that it is an attraction force.
The resultant force F can be found by the formula
F = √Fx2+Fy2 ------------(3)
where Fx = F₁x + F₂x and Fy = F₁y + F₂y
where Fx = F₁x + F₂x and Fy = F₁y + F₂y
Fx = F₁ cos 60° + F₂ cos 60°
Fx = 7.2 N x 0.5 + (-10.80 N ) x 0.5
Fx = 3.6 N + (-5.4 N )
Fx = -1.8 N ------------------(4)
and
Fy = F₁ sin 60° - F₂ sin 60° (opposite in direction)
Fy = 7.2 N x 0.866 - (-10.80 N ) x 0.866
Fy = 6.235 N + 9.353 N
Fy = 15.588 N --------------(5)
Putting Eqn (4)and (5) in eqn (3) we have
F = √Fx2+Fy2
F = √(-1.8N)2+(15.588N)2
F = √3.24N2+242.986N2
F = √246.226N2
F = 15.693 N
F = 15.7 N ------------------------Ans.
Thus the magnitude of the resultant force acting on q₃ is 15.7 N
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