4.15 A magnetic field of 100 G (1 G = 10^–4 T) is required which is uniform in a region of linear dimension about 10 cm and area of cross-section about 10^–3 m². The maximum current-carrying capacity of a given coil of wire is 15 A and the number of turns per unit length that can be wound round a core is at most 1000 turns m^–1. Suggest some appropriate design particulars of a solenoid for the required purpose. Assume the core is not ferromagnetic.

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Payal Gupta | Contributor-Level 10

5 months ago

4.15 Magnetic field strength, B = 100 G = 100 $10^{- 4}$ T

Number of turns per unit length, n = 1000 turns / m

Current flowing in the coil, I = 15 A

$μ_{0}$ = Permeability of free space = 4 $π \times 10^{- 7}$ T m $A^{- 1}$

Magnetic field is given by the relation,

B = $μ_{0} n I$ or nI = $\frac{B}{μ_{0}}$ = $\frac{100 \times 10^{- 4}}{4 π \times 10^{- 7}}$ = 7957.75 A $m^{- 1} \approx 8000$ A $m^{- 1}$

If the length of the coil is taken as 50 cm, radius 4 cm, number of turns 400 and current 10 A, then these values are not unique for the given purpose. There is always a possibility of some adjustments with limits.

4.15 Magnetic field strength, B = 100 G = 100 <math> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </mrow> </msup> </math> TNumber of turns per unit length, n = 1000 turns / mCurrent flowing in the coil, I = 15 A <math> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </math> = Permeability of free space = 4 <math> <mi>π</mi> <mo>×</mo> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>7</mn> </mrow> </mrow> </msup> </math> T m <math> <msup> <mrow> <mrow> <mi>A</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> </math> Magnetic field is given by the relation, B = <math> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> <mi>n</mi> <mi>I</mi> </math> or nI = <math> <mfrac> <mrow> <mrow> <mi>B</mi> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </math> = <math> <mfrac> <mrow> <mrow> <mn>100</mn> <mo>×</mo> <mi mathvariant="normal"> </mi> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> <mi>π</mi> <mo>×</mo> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>7</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </math> = 7957.75 A <math> <msup> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> <mo>≈</mo> <mn>8000</mn> <mi> </mi> </math> A <math> <msup> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> </math> If the length of the coil is taken as 50 cm, radius 4 cm, number of turns 400 and current 10 A, then these values are not unique for the given purpose. There is always a possibility of some adjustments with limits.

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