Question

Two charges of 4 μC and 16 μC are separated by a distance of 30 cm. Calculate the distance of the point from the smaller charge where the electric field intensity between them will be zero

Answer 1

THE POINT O WHERE THE ELECTRIC INTENSITY IS ZERO IS AT A DISTANCE OF 0.14 m FROM POINT CHARGE A.

Step-by-step explanation:

Let the point charges be written as qA and qB

qA = 4 * 10^-6 C

qB = 16 *10^-6 C

Distance (d) between the charges is 30 cm = 0.3 m

Let the point where the electric intensity is 0 be 0

Assume that the distance 0 of point chareg A is x

then, the distance 0 from point charge B is 0.3 - x

Since the electric field imtensity is zero at point O,

the intensity at point O due to qA = the intensity at point O due to qB

E (qA) = E (qB)

I/ 4πEo q / d^2 = 1 /4πEo qB/d^2

I/ 4πEo cancels out from both equation, then we have;

4 * 10^-6 / x^2 = 16 *10^-6 / (0.3-x)^2

4 / x ^2 = 16 / (0.3-x)^2

4/x^2 = 16 / 0.09 - x^2

Cross multiply, we have;

4 * 0.09 - x^2 = 16 * x^2

0.36 = 16 x^2 + x^2

0.36 = 17 x^2

x^2 = 0.36 / 17

x^2 =0.02

x = square root of 0.02

x= 0.14 m

So therefore, the point O is at a distance of 0.14 m from point charge qA.

Answer 2

r = 0.1 m = 10 cm

Step-by-step explanation:

To find the position in which the magnitude of the electric field is zero you use the following formula for E:

`E=k(q)/(r^2)`

you can formulate the total electric field of both charges in the following way:

`E_T=k(q_1)/(r^2)-k(q_2)/((0.3-r)^2)`

But ET = 0 for r, the you have:

`E_T=0\\\\k(q_1)/(r^2)=k(q_2)/((0.3-r)^2)\\\\(0.3-r)^2q_1=q_2r^2\\\\0.09q_1-0.6q_1r-q_1r^2=q_2r^2\\\\(q_2-q_1)r^2+0.6q_1r-0.09q_1=0`

Thus, you have a quadratic equation. You replace the values of the charges and then you use the quadratic formula to get the roots of the polynomial:

`(16*10^(-16)-4*10^(-6)})r^2+(0.6)(4*10^(-6))r-0.09(4*10^(-6))=0\\\\12r^2+2.4r-0.36\\\\r_(1,2)=(-2.4\pm √((2.4)^2-4(12)(-0.36)))/(2(12))\\\\r=0.1m`

when you has taken the positive root because it has physical meaning.

hence, the distance in which ET = 0 is 0.1m = 10cm from the smaller charge