Two point charges, q_1 = +1.6 \times 10^{-8}\text{ C} and q_2 = +6.4 \times 10^{-8}\text{ … — JAMB UTME

Two point charges, $q_1 = +1.6 \times 10^{-8}\text{ C}$ and $q_2 = +6.4 \times 10^{-8}\text{ C}$ , are fixed in a vacuum at a distance of $0.60\text{ m}$ apart. At what distance from $q_1$ along the line joining the two charges is the net electric field intensity equal to zero?

$0.20\text{ m}$ Cevap
B
$0.40\text{ m}$
C
$0.12\text{ m}$
D
$0.30\text{ m}$

Cevap

The distance from

q_1

where the net electric field intensity is zero is

0.20\text{ m}

The net electric field is zero where the magnitudes of the electric fields produced by both charges are equal (

E_1 = E_2

). Setting up

\frac{k q_1}{x^2} = \frac{k q_2}{(0.60 - x)^2}

with

q_2 = 4 q_1

yields

\frac{1}{x^2} = \frac{4}{(0.60 - x)^2}

. Taking the square root gives

\frac{1}{x} = \frac{2}{0.60 - x}

, which yields

x = 0.20\text{ m}

from

q_1

Adım Adım Çözüm

Set up the condition for zero net electric field intensity.

The electric field magnitudes produced by

q_1

and

q_2

at distance

x

from

q_1

must be equal in magnitude and opposite in direction:

E_1 = E_2

Since both charges are positive, the point of zero net electric field must lie on the line segment connecting them.

Substitute the electric field formula into the equilibrium equation.

\frac{k q_1}{x^2} = \frac{k q_2}{(d - x)^2}

, where

d = 0.60\text{ m}

Electric field intensity due to a point charge is given by

E = \frac{k q}{r^2}

Simplify the equation by canceling common terms and substituting known charge values.

\frac{1.6 \times 10^{-8}}{x^2} = \frac{6.4 \times 10^{-8}}{(0.60 - x)^2} \implies \frac{1}{x^2} = \frac{4}{(0.60 - x)^2}

Dividing both sides by

k \times 1.6 \times 10^{-8}

reduces the numerical coefficients to simple integers.

Take the square root of both sides and solve for

x

\frac{1}{x} = \frac{2}{0.60 - x} \implies 0.60 - x = 2x \implies 3x = 0.60 \implies x = 0.20\text{ m}

Taking the square root removes the quadratic terms and gives a linear relation for the distance

x

from

q_1

Anahtar Kavram

Electric field superposition and point of zero field intensity between like point charges

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Cevap

Adım Adım Çözüm

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