A source charge of [tex]\( 3 \mu C \)[/tex] generates an electric field of [tex]\( 2.86 \times 10^5 \, N/C \)[/tex] at the location of a test charge.

What is the distance, to the nearest hundredth, of the test charge from the source charge?

[tex]\( k = 8.99 \times 10^9 \, N \cdot m^2 / C^2 \)[/tex]

[tex]\[ \square \, m \][/tex]



Answer :

To determine the distance between the source charge and the test charge, we can use Coulomb's Law as it pertains to electric fields. The relationship is given by:

[tex]\[ E = k \frac{Q}{r^2} \][/tex]

where:
- [tex]\(E\)[/tex] is the electric field,
- [tex]\(k\)[/tex] is Coulomb's constant ( [tex]\(8.99 \times 10^9 \, \frac{N \cdot m^2}{C^2} \)[/tex] ),
- [tex]\(Q\)[/tex] is the source charge ( [tex]\(3 \, \mu C = 3 \times 10^{-6} \, C\)[/tex] ), and
- [tex]\(r\)[/tex] is the distance we need to find.

We need to solve for [tex]\(r\)[/tex], the distance. Rearranging the equation to solve for [tex]\(r^2\)[/tex], we get:

[tex]\[ r^2 = k \frac{Q}{E} \][/tex]

Taking the square root of both sides, we get:

[tex]\[ r = \sqrt{ \frac{kQ}{E} } \][/tex]

Substituting in the given values:

[tex]\[ r = \sqrt{ \frac{8.99 \times 10^9 \, \frac{N \cdot m^2}{C^2} \times 3 \times 10^{-6} \, C}{2.86 \times 10^5 \, \frac{N}{C}} } \][/tex]

Calculate the value inside the square root:

[tex]\[ r = \sqrt{ \frac{8.99 \times 10^9 \times 3 \times 10^{-6}}{2.86 \times 10^5} } \][/tex]

Divide the products in the numerator and the denominator:

[tex]\[ r = \sqrt{ \frac{26.97 \times 10^3}{2.86 \times 10^5} } \][/tex]

Simplify the exponent part:

[tex]\[ r = \sqrt{ \frac{26.97}{2.86 \times 10^2} } \][/tex]

[tex]\[ r = \sqrt{ \frac{26.97}{286} } \][/tex]

Calculate the fraction:

[tex]\[ r = \sqrt{ 0.09432 } \][/tex]

Finally, take the square root:

[tex]\[ r \approx 0.30708418927176845 \][/tex]

Rounding this to the nearest hundredth:

[tex]\[ r \approx 0.31 \, m \][/tex]

Thus, the distance of the test charge from the source charge, rounded to the nearest hundredth, is:

[tex]\[ \boxed{0.31} \, m \][/tex]