Shouldn't They Be the Same?

Electricity and Magnetism Level 3

Suppose the series-RL circuit pictured above is plugged into a standard residential wall outlet in Tokyo, Japan, and it draws a current of RMS magnitude I T I_T .

Then the same circuit is plugged into a standard residential wall outlet in Osaka, Japan, and it draws a current of RMS magnitude I O I_O .

What is I T I O \large{\frac{I_T}{I_O}} ?

Hint: You may want to consult this reference


The answer is 1.0807.

Steven Chase by Steven Chase , 37, USA

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1 solution

João Areias
Mar 2, 2018

The impedance of the circuit is given by:

Z = 100 + 0.25 ⋅ 2 ⋅ π ⋅ f Z = 100 + 0.25 \cdot 2 \cdot \pi \cdot f

We know that in Tokio the frequency of the outlet is f = 50 H z f = 50Hz while in Osaka f = 60 H z f = 60Hz , replacing this with the formula above we have

Z T ≈ ( 100 + 78.5398 j ) Ω Z_T \approx (100+78.5398j) \Omega

Z O ≈ ( 100 + 94.24778 j ) Ω Z_O \approx (100+94.24778j) \Omega

From I T I O \frac{I_T}{I_O} it follows that:

I T I O = V Z T V Z O = Z O Z T \frac{I_T}{I_O} = \frac{\frac{V}{Z_T}}{\frac{V}{Z_O}} = \frac{Z_O}{Z_T}

I T I O = Z O Z T \frac{I_T}{I_O} = \frac{Z_O}{Z_T}

Now one can substitute the values into the equation, so it follows that:

I T I O ≈ 100 + 78.5398 j 100 + 94.24778 j ≈ 1.076 + 0.09715 j \frac{I_T}{I_O} \approx \frac{100+78.5398j}{100+94.24778j} \approx 1.076+0.09715j

which has a magnitude of approximately 1.0807 1.0807

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