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The turns of solenoid, designed to provi...

The turns of solenoid, designed to provide a given magnetic fulx density along its axis, are wound to fill the space between two concentric cylinder of fixed radii.How should the diameter d of the wire used be chosen so as to minimize the heat dissipated in the winding?

A

Wire should be multiple of 5d

B

Wire should be multiple of `d//3`

C

Wire is independent of d

D

Can 't say

Text Solution

Verified by Experts

The correct Answer is:
C
The cross sectional area of the space to be filled is fiexd withlist that of the wire varies as `d^(2)` Thus `nprop d^(-2)` . The resistacne of one turn is inversely proportional to the cross sectional area of the wire i.e varies as `d^(-2)` and hence the resistance per unit length of the solenoid is `Rpropnd^(-2)propd^(-4)` The flux density `Bis prop nI` and therefore the required current `Ipropn^(-1)propd^(2)` The heat dissipated per unit length is `RI^(2)`, Which `propd^(-4)(d^(2)^(2))` i.e independent of chosen so far as the heatin effect is concerned.
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A long solenoid having n = 200 turns per metre has a circular cross-section of radius a_(1) = 1 cm . A circular conducting loop of radius a_(2) = 4 cm and resistance R = 5 (Omega) encircles the solenoid such that the centre of circular loop coincides with the midpoint of the axial line of the solenoid and they have the same axis as shown in Fig. A current 't' in the solenoid results in magnetic field along its axis with magnitude B = (mu)ni at points well inside the solenoid on its axis. We can neglect the insignificant field outside the solenoid. This results in a magnetic flux (phi)_(B) through the circular loop. If the current in the winding of solenoid is changed, it will also change the magnetic field B = (mu)_(0)ni and hence also the magnetic flux through the circular loop. Obvisouly, it will result in an induced emf or induced electric field in the circular loop and an induced current will appear in the loop. Let current in the winding of solenoid be reduced at a rate of 75 A //sec . Magnetic of induced electric field strength in the circular loop is nearly We know that there is magnetic flux through the circular loop because of the magnetic field of current in the solenoid. For the purpose of circular loop, let us call it the external magnetic field. As current in the solenoid is reducing, external magnetic field for the circular loop also reduced resulting in induced current in the loop. Finally, as the solenoid current becomes zero, external field for the loop also becomes zero and stop changing. However, induced current in the loop will not stop at the instant at which the external field stops changing. This is because induced current itself produces a magnetic field that results in a flux through the loop. External field becoming zero without any further change will compel the induced current in the loop to become zero and so magnetic flux through the loop due to change in induced current will also change resulting in a further induced phenomenon that sustains currents in the loop even after the external field becomes zero.

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Knowledge Check

  • A capacitor can be used to produce a desired electric field. We consldered the parallel plate arrangement as a basic type of capacitor. Similarly, an inductor (symbol ) con be used to near the middle of a long solenoid ) as our basic type fo inductor. if we establish a current i in the winding ( or turns ) of an inductro ( a solenoid ) the current produces a magnetic flux phi_(B) through the central region of the inductor. the inductance of the inductor is then [ L = (N phi_(E))/(i) ] ( inductance defined ) where N is the number of turns. the windings of the inductor are said to the linked by the shared flux, and the product N phi_(E ) is called the magnetic flux linkage . the inductance L is thus a measure of flux produced by the inductor per unit of current . Now, consider a special type of inductor whose radius of turn R as shown in the adjacent figure and total number of turns is N. This special type of solenoid produces a magnetic field |vec(B)| = mu l r^(2) uniformly along the axis of the solenoid as shown in figure. r is the distance from the axis of solenoid. If it is given that total flux linkage through the inductor is KIL^(2) (where L is the inductance). then its inductance is given by

    A
    K
    B
    `K^(2)`
    C
    1/K
    D
    1/`K^(2)`

  • III. Read the following text and answer the following questions on the basis of the same: TOROID A toroid is a coil of insulated or enamelled wire wound on a donut-shaped form made of powdered iron. A toroid is used as an inductor in electronic circuits, especially at low frequencies where comparatively large inductances are necessary. A toroid has more inductance, for a given number of turns, than a solenoid with a core of the same material and similar size. This makes it possible to construct high-inductance coils of reasonable physical size and mass. Toroidal coils of a given inductance can carry more current than solenoidal coils of similar size, because large-diameter wires can be used, and the total amount of wire is less, reducing the resistance. In toroid, all the magnetic flux is contained in the core material. This is because the core has no ends from which flux might leak off. The confinement of the flux prevents external magnetic fields from affecting the magnetic field in the toroid from affecting other components in a circuit. Standard toroidal transformers typically offer a 95% efficiency, while standard laminated transformers typically offer less than a 90% rating. one of the most important differences between a toroidal transformer and a tradition laminated transformer is the absence of gaps. The leakage flux through the gaps contributes to the stray losses in the form of eddy currents (Which is also expelled in the form of heat). A toridal core doesn't have an air gap. The core is tightly wound. The result is a stable predictable toroidal core, free from discontinuities and holes. Audible vibration or hum in transformers is caused by vibration of the windings and core layers from the forces between the coil turns and core laminations. The toroidal transformer's construction helps quiet this noise. in audio, or signal transmitting applications, unwarranted noise will affect sound quality, so a transformer with low audible vibration is ideal. For this reason, many sound system engineers prefer to use toroidal transformer instead of a traditional laminated transformer. Why inductance of solenoid is more than the inductance of a solenoid having same number of turns, core of same material and similar size?

    A
    Core is endless hence there no leakage of flux
    B
    Resistance of wire is less hence magnitude of current flow is more
    C
    Number of turns per unit length is more.
    D
    Both (A) and (B)

  • III. Read the following text and answer the following questions on the basis of the same: TOROID A toroid is a coil of insulated or enamelled wire wound on a donut-shaped form made of powdered iron. A toroid is used as an inductor in electronic circuits, especially at low frequencies where comparatively large inductances are necessary. A toroid has more inductance, for a given number of turns, than a solenoid with a core of the same material and similar size. This makes it possible to construct high-inductance coils of reasonable physical size and mass. Toroidal coils of a given inductance can carry more current than solenoidal coils of similar size, because large-diameter wires can be used, and the total amount of wire is less, reducing the resistance. In toroid, all the magnetic flux is contained in the core material. This is because the core has no ends from which flux might leak off. The confinement of the flux prevents external magnetic fields from affecting the magnetic field in the toroid from affecting other components in a circuit. Standard toroidal transformers typically offer a 95% efficiency, while standard laminated transformers typically offer less than a 90% rating. one of the most important differences between a toroidal transformer and a tradition laminated transformer is the absence of gaps. The leakage flux through the gaps contributes to the stray losses in the form of eddy currents (Which is also expelled in the form of heat). A toridal core doesn't have an air gap. The core is tightly wound. The result is a stable predictable toroidal core, free from discontinuities and holes. Audible vibration or hum in transformers is caused by vibration of the windings and core layers from the forces between the coil turns and core laminations. The toroidal transformer's construction helps quiet this noise. in audio, or signal transmitting applications, unwarranted noise will affect sound quality, so a transformer with low audible vibration is ideal. For this reason, many sound system engineers prefer to use toroidal transformer instead of a traditional laminated transformer. A toroid has _____ inductance, for a given number of turns, than a solenoid with a core of same material and similar size.

    A
    same
    B
    more
    C
    less
    D
    variable
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