8.8 Suppose that the electric field amplitude of an electromagnetic wave is E₀ = 120 N/C and that its frequency is $\mathit{\nu }$ = 50.0 MHz

(a) Determine, B₀, $\mathit{\omega }$ , k, and $\mathit{\lambda }$ .

(b) Find expressions for E and B.

<p><strong>8.8 </strong>Electric field amplitude,&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>E</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub></math></span>&nbsp;= 120 N/C</p><p>Frequency of source,&nbsp;<span title="Click to copy mathml"><math><mi>ν</mi></math></span>&nbsp;= 50 MHz = 50&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>6</mn></mrow></mrow></msup></math></span>&nbsp;Hz</p><p>Speed of light, c = 3&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>8</mn></mrow></mrow></msup></math></span>&nbsp;m/s</p><p>Magnitude of magnetic field strength is given as</p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>B</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub><mo>=</mo><mi></mi></math></span>&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><msub><mrow><mrow><mi>E</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub></mrow></mrow><mrow><mrow><mi>c</mi></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>120</mn></mrow></mrow><mrow><mrow><mn>3</mn><mi mathvariant="normal"></mi><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>8</mn></mrow></mrow></msup></mrow></mrow></mfrac></math></span>&nbsp;= 4&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>7</mn></mrow></mrow></msup></math></span>&nbsp;T = 400&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>9</mn></mrow></mrow></msup></math></span>&nbsp;T = 400 nT</p><p><span title="Click to copy mathml"><math><mi mathvariant="normal">A</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">y</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal"></mi></math></span>&nbsp;<span title="Click to copy mathml"><math><mi>ω</mi></math></span>&nbsp;= 2&nbsp;<span title="Click to copy mathml"><math><mi>π</mi><mi>ν</mi></math></span>&nbsp;= 2&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi>π</mi><mo>×</mo></math></span>&nbsp;50&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>6</mn></mrow></mrow></msup></math></span></p><p>= 3.14&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>8</mn></mrow></mrow></msup></math></span>&nbsp;rad/s</p><p>Propagation constant is given as</p><p>k =&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mi>c</mi></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>3.14</mn><mi mathvariant="normal"></mi><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>8</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mn>3</mn><mi mathvariant="normal"></mi><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>8</mn></mrow></mrow></msup></mrow></mrow></mfrac></math></span>&nbsp;= 1.05 rad/m</p><p>Wavelength of wave is given as</p><p><span title="Click to copy mathml"><math><mi>λ</mi><mo>=</mo><mi></mi><mfrac><mrow><mrow><mi>c</mi></mrow></mrow><mrow><mrow><mi>ν</mi></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>3</mn><mi mathvariant="normal"></mi><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>8</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mn>50</mn><mi mathvariant="normal"></mi><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>6</mn></mrow></mrow></msup></mrow></mrow></mfrac></math></span>&nbsp;= 6.0 m</p><p>Suppose the wave is propagating in the positive x direction. Then, the electric field vector will be in the positive y direction and the magnetic field vector will be in the positive z direction. This is because all three vectors are mutually perpendicular.</p><p>Equation of electric field vector is given as:</p><p><span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>E</mi></mrow></mrow><mo>?</mo></mover></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>E</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>?</mo><mrow><mo>(</mo><mi>k</mi><mi>x</mi><mo>-</mo><mi></mi><mi>ω</mi><mi>t</mi><mo>)</mo></mrow></mrow><mover accent="true"><mrow><mrow><mi>j</mi></mrow></mrow><mo>?</mo></mover></math></span></p><p><span title="Click to copy mathml"><math><mo>=</mo><mi></mi></math></span>&nbsp;120&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>?</mo><mo>(</mo><mn>1.05</mn><mo>×</mo><mi>x</mi><mo>-</mo><mi></mi></math></span>&nbsp;3.14&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>8</mn></mrow></mrow></msup><mi>t</mi><mo>)</mo><mover accent="true"><mrow><mrow><mi>j</mi></mrow></mrow><mo>?</mo></mover></math></span></p><p>Equation of magnetic field vector is given as:</p><p><span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>B</mi></mrow></mrow><mo>?</mo></mover></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>B</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>?</mo><mrow><mo>(</mo><mi>k</mi><mi>x</mi><mo>-</mo><mi></mi><mi>ω</mi><mi>t</mi><mo>)</mo></mrow></mrow><mover accent="true"><mrow><mrow><mi>k</mi></mrow></mrow><mo>?</mo></mover></math></span></p><p>= (400&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi></mi><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>9</mn></mrow></mrow></msup><mo>)</mo><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>?</mo><mrow><mo>(</mo></mrow></mrow><mn>1.05</mn><mo>×</mo><mi>x</mi><mo>-</mo><mi></mi></math></span>&nbsp;3.14&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>8</mn></mrow></mrow></msup><mi>t</mi></math></span>&nbsp;)&nbsp;<span title="Click to copy mathml"><math><mi></mi><mover accent="true"><mrow><mrow><mi>k</mi></mrow></mrow><mo>?</mo></mover></math></span></p>

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