Solve for r e=v+ir
The question provided involves a formula $e = v + ir$, which seems to be a simplified form of Ohm's law. In this context, 'e' represents the electromotive force or voltage across the entire circuit, 'v' is the voltage across a resistor, 'i' is the current flowing through the circuit, and 'r' is the resistance in the circuit. The question is asking you to isolate the variable 'r' and solve for its value in terms of the other variables provided in the equation. This will involve algebraic manipulation to rearrange the terms and solve for the resistance 'r'.
$e = v + i r$
Answer
Solution:
Step 1:
Reformulate the equation to $v + ir = e$.
Step 2:
Isolate $ir$ by subtracting $v$ from both sides, yielding $ir = e - v$.
Step 3:
To solve for $r$, divide the equation $ir = e - v$ by $i$.
Step 3.1:
Apply division to each term: $\frac{ir}{i} = \frac{e}{i} - \frac{v}{i}$.
Step 3.2:
Simplify the equation starting with the left side.
Step 3.2.1:
Eliminate the common $i$ factor.
Step 3.2.1.1:
Remove the $i$ terms: $\frac{\cancel{i} r}{\cancel{i}} = \frac{e}{i} - \frac{v}{i}$.
Step 3.2.1.2:
Simplify to get $r = \frac{e}{i} - \frac{v}{i}$.
Step 3.3:
Now, simplify the right side of the equation.
Step 3.3.1:
Handle each term individually.
Step 3.3.1.1:
Multiply $\frac{e}{i}$ by the complex conjugate of $i$: $r = \frac{e}{i} \cdot \frac{i}{i} - \frac{v}{i}$.
Step 3.3.1.2:
Carry out the multiplication.
Step 3.3.1.2.1:
Combine terms: $r = \frac{ei}{ii} - \frac{v}{i}$.
Step 3.3.1.2.2:
Simplify the denominator.
Step 3.3.1.2.2.1:
Apply exponent rules: $r = \frac{ei}{i^1i} - \frac{v}{i}$.
Step 3.3.1.2.2.2:
Repeat the exponent rule: $r = \frac{ei}{i^1i^1} - \frac{v}{i}$.
Step 3.3.1.2.2.3:
Combine exponents using $a^{m}a^{n} = a^{m+n}$: $r = \frac{ei}{i^{1+1}} - \frac{v}{i}$.
Step 3.3.1.2.2.4:
Add the exponents: $r = \frac{ei}{i^2} - \frac{v}{i}$.
Step 3.3.1.2.2.5:
Replace $i^2$ with $-1$: $r = \frac{ei}{-1} - \frac{v}{i}$.
Step 3.3.1.3:
Move the negative from the denominator: $r = -ei - \frac{v}{i}$.
Step 3.3.1.4:
Rewrite $-ei$ as $-ei$: $r = -ei - \frac{v}{i}$.
Step 3.3.1.5:
Multiply $\frac{-v}{i}$ by the complex conjugate of $i$: $r = -ei - \frac{v}{i} \cdot \frac{i}{i}$.
Step 3.3.1.6:
Perform the multiplication.
Step 3.3.1.6.1:
Combine terms: $r = -ei - \frac{vi}{ii}$.
Step 3.3.1.6.2:
Simplify the denominator.
Step 3.3.1.6.2.1:
Apply exponent rules: $r = -ei - \frac{vi}{i^1i}$.
Step 3.3.1.6.2.2:
Repeat the exponent rule: $r = -ei - \frac{vi}{i^1i^1}$.
Step 3.3.1.6.2.3:
Combine exponents using $a^{m}a^{n} = a^{m+n}$: $r = -ei - \frac{vi}{i^{1+1}}$.
Step 3.3.1.6.2.4:
Add the exponents: $r = -ei - \frac{vi}{i^2}$.
Step 3.3.1.6.2.5:
Replace $i^2$ with $-1$: $r = -ei - \frac{vi}{-1}$.
Step 3.3.1.7:
Recognize that dividing two negatives yields a positive: $r = -ei + \frac{vi}{1}$.
Step 3.3.1.8:
Simplify the division by $1$: $r = -ei + vi$.
Knowledge Notes:
This problem involves solving a linear equation with complex numbers. The key knowledge points include:
Linear Equations: Equations where the highest power of the variable is one.
Complex Numbers: Numbers of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with the property that $i^2 = -1$.
Isolating Variables: The process of manipulating an equation to express one variable in terms of others.
Complex Conjugates: For a complex number $a + bi$, its complex conjugate is $a - bi$. Multiplying a complex number by its conjugate results in a real number.
Exponent Rules: Specifically, the power rule $a^m a^n = a^{m+n}$, which is used to combine like bases with exponents.
Simplifying Expressions: This involves reducing expressions to their simplest form by performing arithmetic operations and combining like terms.
In this problem, we used these concepts to isolate $r$ and then simplify the expression by multiplying by the complex conjugate to remove the imaginary unit from the denominator.
