# AC circuit containing inductive coil of zero ohmic resistance or non inductive ohmic resistance

**The inductive reactance is different from the ohmic resistance where t****he inductive reactance for a coil of zero resistance does not cause loss in electric energy , because the current resistance is due to the reverse induced emf where the electric energy is stored in the coil in the form of a magnetic field , ****The non inductive ohmic resistance causes loss in electric energy in the form of heat energy .**

**AC circuits**

**The value of the self inductance coefficient ( L ) for a solenoid coil is determined by the relation : L = μAN²/l , ****The electric current intensity passing in an induction coil of no resistance is determined from the relation :**

**I = Voltage difference at coil terminals ( V _{L}) / Inductive reactance of the coil ( X_{L})**

**In high frequencies , the inductive reactance ( X _{L }) becomes very big where ( X_{L}∝ f ) , Consequently the electric current intensity decreases where ( I ∝ 1/X_{L} ) and the circuit becomes as if it is opened .**

**AC circuit containing inductive coil of zero ohmic resistance **

**When connecting inductive coil of zero ohmic resistance to AC source and a switch in series , then on closing the circuit , the electric current grows gradually from zero to maximum value at a rate ( ΔI/Δt ) .**

**The variation in the current intensity as time passes generates reverse induced emf by self induction equals ( − L ΔI/Δt ) , ****Where : L = self induction coefficient of the coil .**

**The frequency of the induced emf is the same as that of the AC supply but acting in the opposite direction to the emf of the supply , So , t****he instantaneous value of the potential difference ( V ) = − L ΔI/Δt**

**The phase difference between the current and the potential difference **

**The electric current intensity ( I ) varies with the phase angle according to sine curve as in figure , the value ( ΔI/Δt ) represents the slope of the tangent drawn to the curve where :**

**It reaches its peak value when the phase angle equals zero that means that the voltage ( V ) reaches maximum values .****The slope decreases gradually to reach zero when ( I ) reaches its peak value and the voltage reaches zero .****The slope ( ΔI/Δt ) becomes negative when the current intensity decreases and the voltage becomes negative value .**

**So , the voltage ( V ) leads the current ( I ) by ¼ cycle or by a phase angle 90° due to the self induction of the coil .**

**The inductive reactance in a coil ( X**_{L} )

_{L})

**Since the reverse induced emf produced by self induction in a coil of zero resistance cause some kind of resistance to the flow of the original current , it is called inductive reactance , ****The inductive reactance in a coil ( X _{L}) is the opposition to the flow of the AC current through the coil due to its self-inductance .**

**The inductive reactance is measured in Ohm ( Ω ) , ****The inductive reactance is determined from the relation : X _{L} = 2πfL = ω L , **

**Where : ( L ) coefficient of self induction , ( f ) frequency of the current passing in the coil , ( ω ) the angular velocity .**

**When the inductive reactance of a coil = 100 Ω , It means that the opposition to the flow of the AC current through the coil due to its self induction = 100 Ω .**

**Factors affecting the inductive reactance of a coil **

**Current frequency ( f ) , ( directly proportional ) .****Self induction coefficient ( L ) , directly proportional .**

**The inductive reactance of inductor network**

**When connecting many inductive coils together ( away from each other ) then , ****If conductors are connected in series .**

**L = L _{1 }+ L_{2 }+ L_{2 }**

**X _{L} = ( X_{L} )_{1 }+ ( X_{L} )_{2 }+ ( X_{L} )_{3 }**

**If the self induction coefficients for all coils are equal and number of coils ( n ) .**

**L = n L _{1 , }X_{L} = n ( X_{L} )_{1 }**

**If conductors are connected in parallel .**

**1/L = 1/L _{1 }+ 1/L_{2 }+ 1/L_{2 }**

**1/X _{L} =1/( X_{L} )_{1 }+ 1/( X_{L} )_{2 }+ 1/( X_{L} )_{3}**

**If the self induction coefficients for all coils are equal and number of coils ( n ) .**

**L = L _{1} / n , **

**X**

_{L}= ( X_{L})_{1}/ n**AC circuit containing non inductive ohmic resistance **

**When connecting non inductive ohmic resistance to AC source and a switch in series , then closing the circuit , the potential difference between the terminals of the resistance ( R ) :**

**V = V _{max} sin θ = V_{max} sin ωt**

**Where : V = instantaneous value of potential difference , ****V _{max} = maximum value of potential difference , **

**θ = phase angle ( θ = ωt ) , ω = the angular velocity ( ω = 2πf ) .**

**Based on Ohm’s law , the electric current is determined from the relation :**

**I = V/R = V _{max} sin ωt / R = I_{max} sin ωt**

**The potential difference and the current intensity in an ohmic resistance are having the same phase , thus the current and potential grow together up to a maximum value and drop together to zero , ****The current and potential difference in a resistance without induction are represented by two vectors having the same direction . **

**Properties of the alternating current , Hot wire ammeter uses , cons and pros**

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