Question

A +12 nC charge is located at the origin.

a. What are the electric fields at the positions (x, y) = (5.0 cm, 0 cm), (-5.0 cm, 5.0 cm), and (-5.0 cm, -5.0 cm)? Write each electric field vector in component form.

b. Draw a field diagram showing the electric field vectors at these points.

Short answer

a. Electric field at the position$(5.0cm,0cm)$$E_1=4.32\times {10}^{4}N/C$

Electric field at the position $(-5.0cm,5.0cm)$is $2.2\times {10}^{4}N/C$

Electric field at the position $(-5.0cm,-5.0cm)$is $2.2\times {10}^{4}N/C$

b.

Step-by-step solution

Given information

A $+12nC$charge is located at the origin.

the positions$(x,y)=(5.0cm,0cm),(-5.0cm,5.0cm),and(-5.0cm,-5.0cm)$

Explanation (part a)

Electric field due to a charge $q=kxq/d²$where k is a constant equal to $9x10⁹$, q is given charge and d is distance of point from the charge where field is to be measured. Direction of electric field is towards the force that the charge applies on unit positive charge at the given point.

the electric field at the position$(x₁,y₁)=(5.0cm,0cm)$

here ,localid="1650565880183" $q=-12x10⁻⁹C,d=5cm$

localid="1651599488066" $E_1=-9X10⁹x-12x10⁻⁹/(5x10⁻²)²=4.32x{10}^{4}N/C$

It will act towards the origin along$-xaxis.$

localid="1651599500389" $E_1=-4.32x{10}^4\hat{i}$

the electric field at the positionlocalid="1650565789601" $(x₂,y₂)=(-5.0cm,5cm)$in component form

here ,localid="1650566146337" $q=12x10⁻⁹C,d=7cm$

localid="1651599511779" $E_2=9X10⁹x12x10⁻⁹/(7x10⁻²)²=2.2x{10}^{4}N/C$

It will act towards the origin along$+xaxis.$

localid="1651599521829" $E_2=2.2x10⁴\hat{i}$

electric field at the position localid="1650566088623" $(x₃,y₃)=(-5.0cm,-5.0cm)$in component form

distance between point atlocalid="1650566382735" $(0,0)and(x₃,y₃)=(-5.0cm,-5.0cm)$

localid="1650566404272" $d²=(5²+5²)=50cm²$

the electric field at the position localid="1650566421950" $(x₃,y₃)=(-5.0cm,-5.0cm)$in component form

here ,localid="1650566434631" $q=12x10⁻⁹C,d²=50cm$

localid="1651599545154" $E_2=9X10⁹x12x10⁻⁹/50x10⁻⁴=2.2x10⁴N/C$

Electric field in vector form

localid="1651599642007" $\begin{array}{r}E=-4.32\times {10}^4\cos \theta \hat{i}+4.32\times {10}^4\sin \theta \hat{j}\text{where}\cos \theta =5/\sqrt{12}5\\ \sin \theta =10/\sqrt{125}\\ E=-20.75\times {10}^4/\sqrt{12}5\hat{i}+43.2\times {10}^4/\sqrt{125}\hat{j}\end{array}$

Explanation (part b)

The field diagram showing the electric field vectors at these points are shown below