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VladimirAG [237]
3 years ago
6

Two identical point charges are initially 5.3 cm from each other. If they are released at the same instant from rest, how fast w

ill each be moving when they are very far away from each other? Assume they have identical masses of 1.0 mg.
Mathematics
1 answer:
Zanzabum3 years ago
4 0

Answer:

when they are far away from each other they will move at a velocity of v∞/q = 1.84*10⁶ m/(s*C)

Step-by-step explanation:

if we assume that the charges repel according to a Coulomb's law ( we neglect any relativistic effects) , then the force of repulsion is

F rep = k*q*q/r²

where q= charge

k= Coulomb's constant

r= distance

from Newton's second law :

F rep = m*a= m*dv/dt = m*dv/dr * dr/dt = m*v*dv/dr

k*q*q/r²  = m*v*dv/dr  

k*q*q/(m*r²)   = v*dv/dr

(k*q*q/m) ∫ r⁻² dr= ∫v*dv

-2*k*q*q/m*r = v² - v₁²

for r=r₀ , v=0 ,then

-2*k*q*q/m*r₀ = 0 - v₁²

v₁² = 2*k*q*q/m*r₀

v²=  -k*q*q/(m*r)  + v₁²= 2*k*q*q/m* (1/r₀-1/r)

then

v²=  2*k*q²/m* (1/r₀-1/r)

when the masses are far away from each other r→∞ , 1/r → 0, then

v∞²  =  2*k*q²/(m*r₀)

v∞ = √[2*k*q²/(m*r₀)]

since

k=  8.987*10⁹ N·m²/C² , m= 1 mg = 0.001 kg , r₀= 5.3 cm = 0.053 m

v∞/q = √[[2*k/(m*r₀)] = √[2*8.987*10⁹ N·m²/C²/(0.001 kg*0.053 m)] =  1.84*10⁶ m/(s*C)

v∞/q = 1.84*10⁶ m/(s*C)

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