Question

A European outlet supplies \(220 \mathrm{V}\) (rms) at \(50 \mathrm{Hz}\). How many times per second is the magnitude of the voltage equal to $220 \mathrm{V} ?$

Short answer

Answer: The magnitude of the voltage reaches 220V 100 times per second.

Step-by-step solution

Express the voltage as a function of time

We know that the voltage of a sinusoidal wave can be represented as: $$v(t) = V_m \cdot \sin(\omega t)$$ Where \(V_m\) is the peak voltage, \(\omega\) is the angular frequency and \(t\) is the time. To find the peak voltage, we can use the formula: $$V_\text{rms} = \frac{V_m}{\sqrt{2}}$$ Given \(V_\text{rms} = 220V\), we can find the peak voltage as follows: $$V_m = V_\text{rms} \cdot \sqrt{2} = 220V \cdot \sqrt{2}$$ And the angular frequency can be found using: $$\omega = 2\pi f$$ Where \(f\) is the frequency. For this problem, \(f=50Hz\). This gives us: $$\omega = 2\pi \cdot 50Hz$$ Now we can write our sinusoidal voltage function: $$v(t) = 220\sqrt{2} \cdot \sin(100\pi t)$$

Solve the equation for when the magnitude is 220V

We need to find the points in time when the magnitude of \(v(t)\) equals the given voltage magnitude of \(220V\). We can set up the equation as follows: $$|220\sqrt{2} \cdot \sin(100\pi t)| = 220V$$ To solve the equation, we can divide both sides by \(220V\): $$|\sin(100\pi t)| = \frac{1}{\sqrt{2}}$$ Since sine has a maximum value of 1, it can achieve a value of \(\frac{1}{\sqrt{2}}\) twice in each period. Therefore, the value of \(220V\) is reached twice in each cycle.

Find the total number of times \(220V\) is reached per second

We know that the frequency is \(50Hz\), meaning there are 50 cycles of the signal in a second. Since the voltage reaches \(220V\) twice in one cycle, it will do so \(50 \times 2 = 100\) times in a second. So the magnitude of the voltage is equal to \(220V\) 100 times per second.