Solve E*4pir^2=q/varepsilon_{0}

$$E \cdot 4 \pi r ^ { 2 } = \frac { q } { \varepsilon _ { 0 } }$$

Solve for E

$$\left\{\begin{matrix}E=\frac{q}{4\pi \epsilon _{0}r^{2}}\text{, }&\epsilon _{0}\neq 0\text{ and }r\neq 0\\E\in \mathrm{R}\text{, }&q=0\text{ and }r=0\text{ and }\epsilon _{0}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \epsilon _{0}.

$$E\times 4\pi r^{2}\epsilon _{0}=q$$

The equation is in standard form.

$$4\pi \epsilon _{0}r^{2}E=q$$

Divide both sides by 4\pi r^{2}\epsilon _{0}.

$$\frac{4\pi \epsilon _{0}r^{2}E}{4\pi \epsilon _{0}r^{2}}=\frac{q}{4\pi \epsilon _{0}r^{2}}$$

Dividing by 4\pi r^{2}\epsilon _{0} undoes the multiplication by 4\pi r^{2}\epsilon _{0}.

$$E=\frac{q}{4\pi \epsilon _{0}r^{2}}$$

Solve for q

$$q=4\pi E\epsilon _{0}r^{2}$$

$$\epsilon _{0}\neq 0$$

Solution Steps

Multiply both sides of the equation by \epsilon _{0}.

$$E\times 4\pi r^{2}\epsilon _{0}=q$$

Swap sides so that all variable terms are on the left hand side.

$$q=E\times 4\pi r^{2}\epsilon _{0}$$