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Physics Chapter 12 : Topic 13
Magnetic Flux Linkage
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Magnetic flux linkage of a coil is defined as:
Reasoning: Magnetic flux linkage is the total magnetic flux linked with all turns of a coil. $\Longrightarrow$ $\mathbf{N\Phi}$ where $\mathbf{(N)}$ is the number of turns.
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The mathematical expression for magnetic flux linkage is:
Reasoning: Flux linkage is calculated by multiplying the flux through one turn by the number of turns. $\Longrightarrow$ $\mathbf{Flux\ linkage = N\Phi}$
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A coil has $\mathbf{200}$ turns and each turn has magnetic flux of $\mathbf{0.02\ Wb}$. The flux linkage is:
Reasoning: $\mathbf{N}\mathbf{\Phi = 200\ }\left( \mathbf{0.02} \right)\mathbf{= 4\ Wb}$
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If the number of turns in a coil is doubled while magnetic flux remains constant, flux linkage:
Reasoning: Flux linkage is directly proportional to the number of turns: $\mathbf{N\Phi \propto N}$
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A single-turn coil has magnetic flux of $\mathbf{5\ Wb}$. Its flux linkage is:
Reasoning: For one turn $\Longrightarrow$ $\mathbf{N\Phi = 1}\left( \mathbf{5} \right)\mathbf{= 5\ Wb}$
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Increasing the number of turns in a coil increases flux linkage because:
Reasoning: Each turn links with the same magnetic flux, so total linkage increases with the number of turns.
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A coil has $\mathbf{500}$ turns and flux linkage of $\mathbf{100\ Wb}$. The flux through each turn is:
Reasoning: $\mathbf{\Phi =}$ $\frac{\mathbf{N\Phi}}{\mathbf{N}}$ $\Longrightarrow$ $\mathbf{\Phi =}$ $\frac{\mathbf{100}}{\mathbf{500}}\mathbf{= 0.2\ Wb}$
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A coil with more turns compared to another coil in the same magnetic field will have:
Reasoning: More turns mean more flux linkages. $\Longrightarrow$ $\mathbf{Flux\ linkage = N\Phi}$
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A coil of $\mathbf{100}$ turns experiences a magnetic flux of $\mathbf{0.5\ Wb}$. The flux linkage is:
Reasoning: $\mathbf{N\Phi = 100\ }\left( \mathbf{0.5} \right)\mathbf{= 50\ Wb}$
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Magnetic flux linkage increases when:
Reasoning: Flux linkage depends on both the number of turns and flux per turn.
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A coil has $\mathbf{250}$ turns. The magnetic flux through each turn is $\mathbf{4 \times}\mathbf{10}^{\mathbf{- 3}}\mathbf{\ Wb}$. The flux linkage is:
Reasoning: $\mathbf{N\Phi = 250\ (4 \times}\mathbf{10}^{\mathbf{- 3}}\mathbf{) = 1\ Wb}$
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The unit of magnetic flux linkage is:
Reasoning: Since flux linkage is $\mathbf{N\Phi}$, its unit is also weber $\mathbf{(Wb)}$.
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A coil has zero magnetic flux through each turn. Its flux linkage is:
Reasoning: If $\mathbf{\Phi = 0}$ then: $\mathbf{N\Phi = 0}$.
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Two coils have the same magnetic flux through each turn. Coil $\mathbf{A}$ has $\mathbf{100}$ turns and Coil $\mathbf{B}$ has $\mathbf{500}$ turns. The ratio of flux linkages $\mathbf{A\ :B}$ is:
Reasoning: $\mathbf{N}_{\mathbf{A}}\mathbf{\ :}\mathbf{N}_{\mathbf{B}}\mathbf{= 100\ :500 = 1\ :5}$
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If magnetic flux through a coil becomes three times while the number of turns remains constant, flux linkage:
Reasoning: Flux linkage is directly proportional to magnetic flux. $\Longrightarrow$ $\mathbf{N\Phi \propto \Phi}$
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A coil of $\mathbf{300}$ turns is placed in a magnetic field. If magnetic flux per turn changes from $\mathbf{(0.1\ Wb)}$ to $\mathbf{(0.2\ Wb)}$, the flux linkage:
Reasoning: Flux linkage is proportional to flux per turn.
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The concept of magnetic flux linkage is important in:
Reasoning: A change in flux linkage produces an induced $\mathbf{e.m.f.}$ according to Faraday’s law.
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According to Faraday’s law, induced $\mathbf{emf}$ depends on:
Reasoning: $\mathbf{e.m.f = -}$ $\frac{\mathbf{d(N\Phi)}}{\mathbf{dt}}$According to Faraday’s law of induction, the magnitude of the induced electromotive force $\mathbf{(e.m.f)}$ in a circuit depends directly on the number of turns in the coil and the time rate of change of magnetic flux passing through the circuit.
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A coil has $\mathbf{100}$ turns and area $\mathbf{0.5}\mathbf{\ m}^{\mathbf{2}}$ in a magnetic field of $\mathbf{2\ T}$. The flux linkage is:
Reasoning: Flux: $\mathbf{\Phi = BA = 2\ (0.5) = 1\ Wb}$Flux linkage: $\mathbf{N\Phi = 100\ (1) = 100\ Wb}$
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If a coil is rotated so that magnetic flux through each turn decreases, the flux linkage:
Reasoning: Flux linkage depends on the magnetic flux through each turn.
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A $\mathbf{400}$-turn coil has flux linkage of $\mathbf{80\ Wb}$. The magnetic flux through each turn is:
Reasoning: $\mathbf{\Phi =}$ $\frac{\mathbf{N\Phi}}{\mathbf{N}}\mathbf{=}\frac{\mathbf{80}}{\mathbf{400}}\mathbf{= 0.2\ Wb}$The relationship between total flux linkage, the number of turns, and individual magnetic flux is given by the formula: $\mathbf{\lambda = N\ \ .\ \ \Phi}$● $\mathbf{\lambda =}$ Total flux linkage $\mathbf{= 80\ Wb}$● $\mathbf{N =}$ Number of turns $\mathbf{= 400}$● $\mathbf{\Phi =}$ Magnetic flux per turnRearranging the formula to solve for $\mathbf{\Phi}$.$\mathbf{\Phi =}$ $\frac{\mathbf{\lambda}}{\mathbf{N}}$ $\Longrightarrow$ $\mathbf{\Phi =}$ $\frac{\mathbf{80\ Wb}}{\mathbf{400}}$ $\Longrightarrow$ $\mathbf{\Phi = 0.2\ Wb}$
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If the number of turns is increased without changing magnetic field or coil area, then:
Reasoning: Magnetic flux remains the same, but total linkage increases due to more turns.
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A coil has $\mathbf{50}$ turns and magnetic flux of $\mathbf{0.04\ Wb}$. To obtain flux linkage of $\mathbf{4\ Wb}$, the number of turns required is:
Reasoning: $\mathbf{N =}$ $\frac{\mathbf{N\Phi}}{\mathbf{\Phi}}$ $\Longrightarrow$ $\mathbf{N =}$ $\frac{\mathbf{4}}{\mathbf{0.04}}\mathbf{= 100}$
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Magnetic flux linkage is greater for a coil with:
Reasoning: Both number of turns and flux per turn increase flux linkage.
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A coil has $\mathbf{1000}$ turns. If magnetic flux per turn is $\mathbf{10}^{\mathbf{- 4}}\mathbf{\ Wb}$, the flux linkage is:
Reasoning: $\mathbf{N\Phi = 1000\ }\left( \mathbf{10}^{\mathbf{- 4}} \right)\mathbf{= 0.1\ Wb}$
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The difference between magnetic flux and magnetic flux linkage is:
Reasoning: Magnetic flux is through one turn, while linkage is the combined effect of all turns.
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If a coil has $\mathbf{500}$ turns and each turn links the same flux, the total linkage is:
Reasoning: Total linkage is $\mathbf{N\Phi}$, which is larger for multiple turns.
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A coil’s flux linkage changes from $\mathbf{20\ Wb}$ to $\mathbf{60\ Wb}$. The change in flux linkage is:
Reasoning: $\mathbf{\Delta}\left( \mathbf{N\Phi} \right)\mathbf{= 60 - 20 = 40\ Wb}$
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A larger flux linkage generally produces a larger induced $\mathbf{emf}$ when:
Reasoning: Faraday’s law depends on the rate of change of flux linkage.
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The main purpose of using many turns in a coil is to:
Reasoning: More turns increase $\mathbf{N\Phi}$, resulting in greater induced $\mathbf{e.m.f.}$ when flux changes.
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