Impedance of Free Space: Formula, Derivation and Value

Impedance of free space is the resistance of a plane wave traveling in free space. Simply, impedance refers to resistance. Mathematically, it is written as Z₀ and it is calculated by the square root of the ratio of permeability of free space to the permittivity of free space. Ultimately, it is the relation between the magnetic field and electric field in the electromagnetic wave.

Key Terms: Impedance, Resistance, Space, Electromagnetic Wave, Electric Field, Permittivity, Universal Constants, Ohm

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Value of Impedance of Free Space

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As the permittivity of free space and permeability of free space both are universal constants, the value of Z₀ is also a universal constant and its value is 

It has the same unit as resistance i.e. Ohm. Impedance of free space can be also defined as the relation between the magnetic field and electric field in the electromagnetic wave.

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Mathematical Formula of Impedance of Free Space

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Mathematically, the impedance of free space can be represented as:

⇒ Z₀ = |E|/|H| 

= √(u_o/ε_o) 

= u_oc_o 

= 1/ε_oc_o

Where,

• |E| = electric field strength
• |H| = magnetic field strength
• u_o = permittivity of free space
• ε_o = permeability of free space
• C_o = speed of light

It is also sometimes called an Admittance of Free Space.

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Characteristic of Impedance of Free Space

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The value of the impedance of free space is almost the same as that of dry air because dry air has very less effect on the magnitude of permeability and permittivity.

• Impedance is the property of the medium in which the electromagnetic wave is travelling despite its association with any kind of power source or load at any end.
• Impedance is found to be independent of the length or distance to which the wave is travelling.This means the characteristic impedance of the universe is equivalent to the characteristic impedance of a small circuit in free space.
• Impedance of free space is the same as the ratio of the penetrability of free space (henry per meter) to the penetrability of free space(farad per meter). It is also found to be 377 ohms.
• Impedance is the property of any medium that can allow the transmission of an electromagnetic wave irrespective of the fact that it is associated with a power source at one terminal and a load at the other end or not.

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Impedance of Free Space Derivation

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The impedance of free space is represented by Z₀. The wave impedance in free space is given by:

Z₀ = E/H= √ u_oc_o  

c₀ = 100 = 299792458 m/s 

Z₀ = E/H = \(\frac{1}{\sqrt{ \mu_o \epsilon_o}}\) = 4 × 10^-7 H/m × 299792458 m/s

= 376.73

Why does the vacuum have a non-zero characteristic impedance towards electromagnetic radiation?

The impedance of free space is defined as the square root of the ratio of the permittivity of free space to the permeability of free space. This value is approximately 376 ohms.

Now, impedance is known to be a form of resistance. The vacuum is considered to contain nothing, absolute emptiness. But how can electromagnetic waves which require a medium to propagate, can travel through a vacuum. That means free space is not actually empty because we have defined permeability and permittivity for it.

Impedance Plane

About 99.99% universe is empty space. It has been predicted that when the milky way and Andromeda galaxy(the galaxy nearest to our galaxy) will collide after approximately 4 billion years, none of the stars in either of the galaxy will collide!

Though there will be no collisions, both the galaxy will surely be affected by the massive gravitational interactions among them. The universe has been created with much more amount of stability than we actually think. 

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Previous Year Questions

1. An aluminium rod and a copper rod are taken … [DUET 2007]
2. In the adjacent shown circuit, a voltmeter of internal resistance … [BITSAT 2009]
3. A wire of resistance 12 ohm is in the form of a circle … [JIPMER 1999]
4. A cell of constant emf first connected to a resistance … [BITSAT 2006]
5. In an a.c. circuit the e.m.f. … [NEET 2008]
6. A circuit contain two resistors R1 and R2 in series … [JIPMER 2019]
7. The masses of three copper wires are in the ratio … [VITEEE 2015]
8. A letter ' A ' is constructed of a uniform wire with resistance … [BITSAT 2012]
9. In a region of magnetic induction … [NEET 1998]
10. A 10V battery with internal resistance … [JKCET 2018]  
11. Resistances 1 Ω,2 Ω and 3 Ω are connected to form a triangle … [VITEEE 2017]
12. A milliammeter of range 10 mA has a coil of resistance … [KCET 2001]
13. Twelve resistors each of resistance  … [BITSAT 2019]
14. The equivalent resistance between the points P and Q in the network … [UPSEE 2006]
15. What is the potential drop between points A and C  … [BCECE 2006]
16. A cell has terminal voltage … [JEE Advanced 2016]

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Things to Remember

• Impedance is another term for resistance.
• The impedance of free space is also sometimes called the admittance of free space.
• The impedance of free space is calculated to be 376 ohms.
• Mathematically, it is expressed as Z₀ and it is the square root of the ratio of permeability of free space to the permittivity of free space.
• The value of the impedance of free space is almost the same as that of dry air because dry air has very less effect on the magnitude of permeability and permittivity.
• Characteristic impedance is found to be independent of the length or distance to which the wave is traveling.
• The characteristic impedance of the universe is equivalent to the characteristic impedance of a small circuit in free space.

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