Question

A particle with charge Q is on the y axis a distance a from the origin and a particle with charge q is on the x axis a distance d from the origin. The value of d for which the x component of the force on the second particle is the greatest is?

Answer 1

Answer:

Explanation:

Force on q due to Q

= k Qq / ( a² + d²)

x component

= k Qq / ( a² + d²)  x d /√ ( a² + d²)

F = kQq d/ ( a² + d²)³/²

differentiating F with respect to d

dF / Dd = kQq  [ d. -3/2 ( a² + d²)⁻⁵/² 2d + ( a² + d²)⁻³/²]=0 for maximum F

- 3d² / ( a² + d²) + 1 = 0

a² + d² = 3 d²

a² = 2d²

d = a / √2

Answer 2

The value of d for which the x component of the force on the second particle is the greatest is; d = (√2)a

The x-component of the force is given as;

F = KQq(cos θ)/(4a² + d²)

where;

cos θ = d/√((4a² + d²)

This means;

F = KQq(d/√((4a² + d²))/(4a² + d²)

⇒ F = kQqd/(4a² + d²)^(³/₂)

We will now differentiate with respect to d and equate to zero, from online calculator for differentiation, we will get;

d = (√2)a

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