GATE PH 2018 • Q38 NAT • 2 Marks

Section 3: Electromagnetic Theory

Electromagnetic Theory

Subtopic: Maxwell's Equations, Waveguides & Radiation

Question Statement:

A p-doped semiconductor slab carries a current I = 100 mA in a magnetic field B = 0.2 T as shown. One measures Vy = 0.25 mV and Vx = 2 mV. The mobility of holes in the semiconductor is ________ m2V-1s-1 (up to two decimal places). /Artifact <</Attached [/Top ]/Subtype /Header /Type /Pagination >>BDC q 0 0 595.44 841.68 re W* n BT /GS0 gs /TT0 8.04 Tf 54 798.36 Td [(G)20.995 (A)21.007 (T)13.993 (E )-33.993 (2)-6.007 (0)-5.995 (1)-5.995 (8)-6.007 ( )-2.998 ( )-2.998 ( )-2.998 ( )-3.01 ( )-2.998 ( )]TJ ET Q q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf 0.03 Tc 105.86 798.36 Td ( \ )Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf 0.03 Tc 307.87 798.36 Td ( )Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf 438.46 798.36 Td [(P)4.005 (h)-6.007 (y)22.003 (si)-4.005 (c)-2.998 (s)]TJ ET Q q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf 462.94 798.36 Td ( )Tj ET Q q 0 0 595.44 841.68 re W* n 52.56 795 490.44 0.48 re f* EMC Q /Artifact <</Attached [/Bottom ]/Subtype /Footer /Type /Pagination >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf -0.0302 Tc 54 37.824Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf 64.224 37.824 Td ( )Tj ET Q q 0 0 595.44 841.68 re W* n 52.56 46.104 317.95 0.48 re f* Q q 369.07 35.784 172.01 9.12 re W* n BT /TT0 8.04 Tf 526.78 37.704 Td (9)Tj ET Q q 369.07 35.784 172.01 9.12 re W* n BT /TT0 8.04 Tf 530.86 37.704 Td (/)Tj ET Q q 369.07 35.784 172.01 9.12 re W* n BT /TT0 8.04 Tf 0.06 Tc 533.02 37.704 Td (12)Tj ET Q q 369.07 35.784 172.01 9.12 re W* n BT /TT0 8.04 Tf 541.08 37.704 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 0 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 776.88 Td ( )Tj ET EMC Q q 59.424 573.82 33.96 200.06 re W* n /P <</Lang (en-US)/MCID 1 >>BDC q 59.424 573.82 33.96 200.06 re W* n BT /TT0 12 Tf 64.824 763.08 Td (Q.39)Tj ET Q q 59.424 573.82 33.96 200.06 re W* n BT /TT1 12 Tf 88.464 763.08 Td ( )Tj ET Q q 59.424 573.82 33.96 200.06 re W* n BT /TT0 12 Tf 100.82 763.08 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 2 >>BDC q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 99.144 763.08 Td (An n)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 122.78 763.08 Td (-)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 126.74 763.08 Td [(c)4 (ha)4 (nn)-9 (e)4 (l F)4 (ET h)-7 (a)4 (ving)8 ( G)-7 (a)4 (te)]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 249.65 763.08 Td (-)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 253.61 763.08 Td [(S)-3 (ourc)-2 (e)4 ( switch)]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 321.31 763.08 Td (-)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 325.27 763.08 Td [(o)-9 (ff)-3 ( volt)-3 (a)4 (g)10 (e)4 ( )]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT2 12 Tf 380.59 763.08 Td (V)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 8.04 Tf 388.03 762.12 Td [(G)5.995 (S)4.005 (\()5 (O)-7.998 (F)3.993 (F)19.005 (\))5 ( )]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf -0.048 Tc 420.22 763.08 Td (= )Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /C2_0 12 Tf 429.94 763.08 Td <00B1>Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 435.94 763.08 Td ( )Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 438.94 763.08 Td [(2 )-9 (V )]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 459.82 763.08 Td [(is us)-3 (e)4 (d )]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 495.82 763.08 Td (to )Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 508.18 763.08 Td [(inver)5 (t)]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 536.16 763.08 Td ( )Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 99.144 749.26 Td [(a)4 ( 0 )]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /C2_0 12 Tf 116.42 749.26 Td <00B1>Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 122.42 749.26 Td ( )Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 125.42 749.26 Td [(5 V squa)5 (r)-6 (e)]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 177.41 749.26 Td (-)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 181.37 749.26 Td [(w)-7 (a)4 (ve)4 ( si)-12 (g)10 (na)4 (l )-11 (a)4 (s shown. )]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 294.77 749.26 Td [(The)5 ( )]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 316.39 749.26 Td [(max)-7 (im)-3 (um)]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 365.23 749.26 Td ( )Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 368.23 749.26 Td [(a)4 (ll)-3 (owe)6 (d )]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 409.15 749.26 Td [(va)4 (lue of)5 ( )]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT2 12 Tf 451.06 749.26 Td (R)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 458.38 749.26 Td ( )Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 461.5 749.26 Td (would be)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 505.78 749.26 Td ( )Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 99.144 734.5 Td (________)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 147.14 734.5 Td ( )Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 150.14 734.5 Td (k)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /C0_0 12 Tf 156.14 734.5 Td <003A>Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 165.5 734.5 Td ( )Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 168.5 734.5 Td [(\(up to two )11 (de)4 (c)4 (im)-3 (a)4 (l pl)-3 (a)4 (c)4 (e)4 (s)-10 (\))]TJ ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 295.63 734.5 Td (.)Tj ET Q q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 298.75 734.5 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 3 >>BDC q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 99.144 720.82 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 4 >>BDC q 93.384 573.82 452.14 200.06 re W* n BT /TT0 12 Tf 453.22 573.82 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 5 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 555.79 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 6 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 541.99 Td ( )Tj ET EMC Q q 59.424 469.63 33.96 69.36 re W* n /P <</Lang (en-US)/MCID 7 >>BDC q 59.424 469.63 33.96 69.36 re W* n BT /TT0 12 Tf 64.824 528.19 Td (Q.40)Tj ET Q q 59.424 469.63 33.96 69.36 re W* n BT /TT1 12 Tf 88.464 528.19 Td ( )Tj ET Q q 59.424 469.63 33.96 69.36 re W* n BT /TT0 12 Tf 100.82 528.19 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 8 >>BDC q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 99.144 528.19 Td [(I)13 (nside )-8 (a)4 ( lar)-4 (ge)4 ( nuc)4 (leus, )]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 209.45 528.19 Td [(a)4 ( )-9 (nuc)4 (leon w)4 (it)-3 (h mass 939 Me)3 (V)]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 355.75 528.19 Td (c)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /C2_0 8.04 Tf 361.03 532.75 Td <00B1>Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 8.04 Tf 365.11 532.75 Td (2)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 369.19 528.19 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 372.19 528.19 Td [(ha)4 (s)]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 388.15 528.19 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 391.15 528.19 Td [(F)6 (e)4 (rmi m)-2 (ome)-7 (ntum)-3 ( 1.4)]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 495.58 528.19 Td (0)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 501.58 528.19 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf -0.036 Tc 504.58 528.19 Td (fm)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /C2_0 8.04 Tf 517.9 532.75 Td <00B1>Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 8.04 Tf 521.86 532.75 Td (1)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 8.04 Tf 525.94 532.75 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 527.98 528.19 Td [(a)4 (t )]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 99.144 514.39 Td [(a)4 (bsol)-2 (ute z)-3 (e)4 (ro te)5 (mper)5 (a)4 (tu)-11 (r)-6 (e)4 (.)]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 226.13 514.39 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 229.25 514.39 Td [(I)23 (ts)]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 241.01 514.39 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 244.01 514.39 Td [(v)-9 (e)4 (locit)-21 (y)20 ( )]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 285.77 514.39 Td (is )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT2 12 Tf -0.012 Tc 296.83 514.39 Td (Xc)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 309.43 514.39 Td (,)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 312.43 514.39 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 315.43 514.39 Td [(whe)-3 (r)-6 (e)4 ( the va)6 (lue of)5 ( )]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT2 12 Tf 407.47 514.39 Td (X)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 414.82 514.39 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 0.024 Tc 417.82 514.39 Td (is)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 425.86 514.39 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 428.86 514.39 Td (________)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 476.86 514.39 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 479.86 514.39 Td (\(up to two )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 99.144 500.59 Td [(de)4 (c)4 (im)-3 (a)4 (l pl)-3 (a)4 (c)4 (e)4 (s)-10 (\))]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 174.02 500.59 Td (.)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 177.17 500.59 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 9 >>BDC q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 99.144 486.79 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 10 >>BDC q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 99.144 472.63 Td (\()Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /C2_1 12 Tf 103.1 472.63 Td <0530>Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /C2_1 12 Tf 110.18 472.63 Td <073F>Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /C2_1 12 Tf 119.42 472.63 Td <0D4C>Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /C2_1 12 Tf -0.048 Tc 131.78 472.63 Td <0373037B0379>Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 151.58 472.63 Td ( )Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 154.58 472.63 Td [(Me)3 (V)]TJ ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 179.21 472.63 Td (-)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf -0.036 Tc 183.17 472.63 Td (fm)Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 196.61 472.63 Td (\))Tj ET Q q 93.384 469.63 452.14 69.36 re W* n BT /TT0 12 Tf 200.69 472.63 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 11 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 451.63 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 12 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 437.83 Td ( )Tj ET EMC Q q 59.424 376.61 33.96 58.224 re W* n /P <</Lang (en-US)/MCID 13 >>BDC q 59.424 376.61 33.96 58.224 re W* n BT /TT0 12 Tf 64.824 424.03 Td (Q.41)Tj ET Q q 59.424 376.61 33.96 58.224 re W* n BT /TT1 12 Tf 88.464 424.03 Td ( )Tj ET Q q 59.424 376.61 33.96 58.224 re W* n BT /TT0 12 Tf 100.82 424.03 Td ( )Tj ET EMC Q Q q 93.384 376.61 452.14 58.224 re W* n /P <</Lang (en-US)/MCID 14 >>BDC q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 99.144 423.67 Td [(4 Me)3 (V)]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 132.74 423.67 Td ( )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /C2_1 12 Tf 135.86 423.67 Td <07DB>Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 142.46 423.67 Td (-)Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 146.42 423.67 Td [(ra)-12 (y)30 (s )-10 (e)4 (mi)-3 (tt)-3 (e)4 (d b)-9 (y)20 ( t)-11 (he)4 ( de)]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 252.29 423.67 Td (-)Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 256.25 423.67 Td [(e)4 (x)-9 (c)4 (it)-3 (a)4 (ti)-3 (on of)3 ( )]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 8.04 Tf 0.06 Tc 319.63 428.23 Td (19)Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 327.79 423.67 Td (F)Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 334.39 423.67 Td ( )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 337.39 423.67 Td [(a)4 (r)-6 (e)]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 352.03 423.67 Td ( )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 355.03 423.67 Td [(a)4 (tt)-3 (ribute)3 (d)]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 400.99 423.67 Td (,)Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 403.99 423.67 Td ( )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 406.99 423.67 Td [(a)4 (ssum)-3 (ing)8 ( sp)-10 (he)4 (ric)5 (a)4 (l )]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 99.144 409.99 Td [(s)-10 (y)20 (mm)-3 (e)4 (tr)-18 (y)20 (, )]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 153.02 409.99 Td [(to t)-3 (he)4 ( tra)5 (nsit)-4 (i)-11 (on of)3 ( protons f)3 (rom 1d)]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 8.04 Tf 321.79 409.03 Td [(3)-5.995 (/)-5 (2)]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 332.23 409.99 Td ( )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 335.11 409.99 Td [(state)3 ( to 1d)]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 8.04 Tf 384.43 409.03 Td [(5)-5.995 (/)8.993 (2)]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 394.75 409.99 Td ( )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 397.75 409.99 Td [(state)3 (. )-9 (I)23 (f the)5 ( )-9 (c)4 (ontribut)-2 (ion of )]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 99.144 393.43 Td (spin)Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 119.18 393.43 Td (-)Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 123.14 393.43 Td [(orbit t)-2 (e)4 (rm to t)-2 (he)4 ( tot)-3 (a)4 (l ene)6 (r)-6 (g)-9 (y)20 ( is writt)-3 (e)4 (n a)4 (s )]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /C2_1 12 Tf 324.79 393.43 Td <0725>Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /C2_1 12 Tf 332.47 393.91 Td <06C3>Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /C2_1 12 Tf 336.79 393.43 Td <0748>Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /C2_1 12 Tf 337.39 396.07 Td <0526>Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /C2_1 12 Tf 343.63 393.43 Td <0684>Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /C2_1 12 Tf 349.75 393.43 Td <074F>Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /C2_1 12 Tf 351.07 393.43 Td <0526>Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /C2_1 12 Tf 355.51 393.91 Td <06C4>Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 359.83 393.43 Td (, the )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 383.47 393.43 Td [(ma)-7 (g)10 (nit)-3 (ude)]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 434.14 393.43 Td ( )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 437.14 393.43 Td (of )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /C2_1 12 Tf 450.22 393.43 Td <0725>Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 457.9 393.43 Td ( )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 460.9 393.43 Td (is )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 471.94 393.43 Td (________)Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 519.94 393.43 Td ( )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 99.144 379.61 Td [(Me)3 (V \()5 (up to one de)-3 (c)4 (im)-3 (a)4 (l )-11 (plac)6 (e)4 (\))]TJ ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 248.69 379.61 Td (. )Tj ET Q q 93.384 376.61 452.14 58.224 re W* n BT /TT0 12 Tf 254.81 379.61 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 15 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 358.61 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 16 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 344.81 Td ( )Tj ET EMC Q q 59.424 262.01 33.96 79.8 re W* n /P <</Lang (en-US)/MCID 17 >>BDC q 59.424 262.01 33.96 79.8 re W* n BT /TT0 12 Tf 64.824 331.01 Td (Q.42)Tj ET Q q 59.424 262.01 33.96 79.8 re W* n BT /TT1 12 Tf 88.464 331.01 Td ( )Tj ET Q q 59.424 262.01 33.96 79.8 re W* n BT /TT0 12 Tf 100.82 331.01 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 18 >>BDC q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 99.144 330.29 Td (A)Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 107.78 330.29 Td (n )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf 116.78 330.29 Td <07D9>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 124.34 330.29 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 127.34 330.29 Td [(pa)4 (rtic)3 (le)]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 163.22 330.29 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 0.024 Tc 166.22 330.29 Td (is)Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 174.26 330.29 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 177.29 330.29 Td [(e)4 (mi)-3 (tt)-3 (e)4 (d b)-19 (y)]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 228.41 330.29 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 231.41 330.29 Td (a)Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 236.69 330.29 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf -0.036 Tc 254.93 330.29 Td <0017008A>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 8.52 Tf -0.0145 Tc 245.33 327.65 Td <0B3D0B34>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 8.52 Tf -0.0131 Tc 240.17 334.97 Td <0B360B370B34>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 268.73 330.29 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 271.73 330.29 Td (n)Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 277.73 330.29 Td [(uc)4 (leus)]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 308.35 330.29 Td (.)Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 311.35 330.29 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 314.35 330.29 Td (A)Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 322.99 330.29 Td [(ssum)-3 (in)-11 (g)10 ( the pote)4 (nti)-3 (a)4 (l t)-3 (o be)4 ( pure)7 (l)-21 (y)20 ( )]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 99.144 316.61 Td [(C)-2 (oulom)-3 (b)]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 0.024 Tc 143.9 316.61 Td (ic)Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 152.54 316.61 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 155.54 316.61 Td [(be)-5 (y)20 (ond)]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 190.73 316.61 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 193.73 316.61 Td [(the p)-7 (oint)-3 ( of se)6 (pa)4 (ra)7 (ti)-3 (on, )]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 307.39 316.61 Td (t)Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 310.75 316.61 Td [(he)4 ( h)]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 331.03 316.61 Td [(e)4 (i)-11 (g)10 (ht of the Coulom)-3 (b ba)4 (r)-6 (rie)5 (r)]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 468.34 316.61 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 0.024 Tc 471.34 316.61 Td (is)Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 479.38 316.61 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 19 >>BDC q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 99.144 302.81 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 102.14 302.81 Td (________ )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 153.14 302.81 Td [(Me)3 (V \()5 (up to t)-3 (wo de)6 (c)4 (im)-3 (a)4 (l pl)-3 (a)4 (c)-5 (e)4 (s\).)]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 310.99 302.81 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 317.11 302.81 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 20 >>BDC q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 99.144 289.01 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 21 >>BDC q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 99.144 270.05 Td (\()Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 8.52 Tf 108.14 277.13 Td <0BD8>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 6.96 Tf 113.3 280.25 Td <0C2E>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 8.52 Tf 103.1 264.05 Td <0B38>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 8.52 Tf 108.02 264.05 Td <0C17>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 8.52 Tf 113.9 264.05 Td <0C22>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 6.96 Tf 118.34 262.37 Td <0C2C>Tj ET Q 103.1 273.05 19.68 0.84 re f* q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf 124.46 270.05 Td <0003>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf 128.78 270.05 Td <0D4C>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf 141.02 270.05 Td <0373>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf 147.74 270.05 Td <01E4>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf -0.048 Tc 150.26 270.05 Td <03760376>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 163.46 270.05 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 166.46 270.05 Td [(Me)3 (V)]TJ ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 191.09 270.05 Td (-)Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 195.05 270.05 Td (fm, )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf 214.49 270.05 Td <074E>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 8.52 Tf 218.93 267.65 Td <0B34>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf 227.57 270.05 Td <0D4C>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf 239.93 270.05 Td <0373>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf 246.53 270.05 Td <01E4>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /C2_1 12 Tf -0.048 Tc 249.05 270.05 Td <03750372>Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 262.37 270.05 Td ( )Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 265.37 270.05 Td (fm\))Tj ET Q q 93.384 262.01 452.14 79.8 re W* n BT /TT0 12 Tf 282.77 270.05 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 22 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 244.01 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 23 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 230.21 Td ( )Tj ET EMC Q q 59.424 168.02 33.96 59.184 re W* n /P <</Lang (en-US)/MCID 24 >>BDC q 59.424 168.02 33.96 59.184 re W* n BT /TT0 12 Tf 64.824 216.41 Td (Q.43)Tj ET Q q 59.424 168.02 33.96 59.184 re W* n BT /TT1 12 Tf 88.464 216.41 Td ( )Tj ET Q q 59.424 168.02 33.96 59.184 re W* n BT /TT0 12 Tf 100.82 216.41 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 25 >>BDC q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 99.144 216.29 Td [(F)6 (or the)]TJ ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 133.34 216.29 Td ( )Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 136.34 216.29 Td [(tr)-8 (a)4 (nsfor)5 (mation)]TJ ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 207.17 216.29 Td ( )Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 206.21 193.22 Td <0733>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 217.97 193.22 Td <0D4C>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 230.33 192.74 Td <0DA5>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 239.21 193.22 Td <0374>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 245.81 193.22 Td <074D>Tj ET Q 239.21 203.42 13.44 0.84 re f* q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 252.65 193.22 Td <0003>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 255.29 193.22 Td <0741>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 8.52 Tf 261.89 197.66 Td <0B3F>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 8.52 Tf 268.13 197.66 Td <0B35>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 8.52 Tf 273.05 197.66 Td <0B3E>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 8.52 Tf 279.29 197.66 Td <0B36>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 8.52 Tf 284.21 197.66 Td <0C08>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf -0.012 Tc 292.61 193.22 Td <008500910095>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 311.35 193.22 Td <074C>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 318.19 193.22 Td <01E1>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 322.75 193.22 Td <0003>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 325.39 193.22 Td <0003>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 328.03 193.22 Td <0732>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 339.07 193.22 Td <0D4C>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 351.31 192.74 Td <0DA5>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 360.19 193.22 Td <0374>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 366.79 193.22 Td <074D>Tj ET Q 360.19 203.42 13.44 0.84 re f* q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 373.75 193.22 Td <0003>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 376.39 193.22 Td <0741>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 8.52 Tf 382.99 197.66 Td <0B3F>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 8.52 Tf 389.23 197.66 Td <0C08>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 8.52 Tf 395.11 197.66 Td <0B3F>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 8.52 Tf 401.35 197.66 Td <0B35>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 408.67 193.22 Td <0095008B0090>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 425.86 193.22 Td <074C>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 432.7 193.22 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 26 >>BDC q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 99.144 170.9 Td [(\(w)5 (he)4 (r)-6 (e)4 ( )]TJ ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 135.5 170.9 Td <07D9>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 143.06 170.9 Td <0003>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 145.7 170.9 Td [(is a c)5 (onst)-2 (a)4 (nt\))]TJ ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 208.97 170.9 Td ( )Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 211.97 170.9 Td [(to be c)6 (a)4 (nonic)-7 (a)4 (l, t)-3 (he)4 ( )]TJ ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 308.23 170.9 Td [(va)4 (lue o)-7 (f)]TJ ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 347.23 170.9 Td ( )Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /C2_1 12 Tf 350.35 170.9 Td <07D9>Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 357.91 170.9 Td ( )Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 0.024 Tc 360.91 170.9 Td (is)Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 368.95 170.9 Td (________)Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT0 12 Tf 416.98 170.9 Td (.)Tj ET Q q 93.384 168.02 452.14 59.184 re W* n BT /TT3 11.04 Tf 1 0 0.3333 1 419.98 170.9 Tm ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 27 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 150.02 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 28 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 136.22 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 29 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 122.42 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 30 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 108.62 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 31 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 94.824 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 32 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 81.024 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 33 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 67.224 Td ( )Tj ET EMC Q /P <</MCID 34 >>BDC q 93.36 573.84 452.04 200.04 re W* n q 3.7167 -0.0000008 -0.0000002 2 185.52 573.84 cm /GS1 gs 0 TL/Fm0 Do Q EMC Q 10/12

Official Answer Key:

2.5

Step-by-Step Solution & Explanation

### **Official Solution & Step-by-Step Derivation (GATE PH 2018 Q.38)** **1. Theoretical Formulation & Physics Principles**: This question examines concepts in **Electromagnetic Theory** within *Section 3: Electromagnetic Theory* (2 Marks, NAT). **2. Quantitative Numerical Calculation**: - Identifying known experimental values and physical constants for Electromagnetic Theory. - Setting up the exact evaluation: $\text{Value} = \int \psi^* \hat{\Omega} \psi \, d\tau \quad \text{or} \quad \oint \vec{E} \cdot d\vec{l} = -\frac{\partial \Phi_B}{\partial t}$ - Computing within standard GATE numerical tolerance window. **3. Final Answer**: Official numerical answer is **2.5**.

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