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QJEEM23Ch8b66: JEE Main 2023 Chemistry - Class 11 JEEM23Ch8b66 Solution: Electrons in a cathode ray tube have been emitted with a…

Electrons in a cathode ray tube have been emitted with a velocity of 1000 m s−1. The number of following statements which is/are true― about the emitted radiation is ____________.

Given : h=6×10−34 J s,me=9×10−31 kg.

(A) The de-Broglie wavelength of the electron emitted is 666.67 nm.

(B) The characteristic of electrons emitted depend upon the material of the electrodes of the cathode ray tube.

(C) The cathode rays start from cathode and move towards anode.

(D) The nature of the emitted electrons depends on the nature of the gas present in cathode ray tube.

Step-by-Step Verified Solution

The number of true statements about the emitted radiation is 2, i.e. statements (A) and (C).

(A) is True: The de-Broglie wavelength of an electron can be calculated as lambda = h / p, where h is Planck's constant and p is the momentum of the electron. The momentum of an electron with a velocity of 1000 m/s is mv, where m is the mass of an electron. Plugging in the values, we get

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padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D706 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c68"></mjx-c></mjx-mi></mjx-texatom></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c76"></mjx-c></mjx-mi></mjx-texatom></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.393em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msup></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.393em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.413em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c39"></mjx-c></mjx-mn></mjx-texatom></mjx-script></mjx-msup><mjx-texatom texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c></mjx-mi></mjx-texatom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-texatom texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-n"><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c6D"></mjx-c></mjx-mi></mjx-texatom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd></mtd><mtd><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">V</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">e</mi></mrow></mrow></msub><mo>=</mo><mn>1000</mn><mrow data-mjx-texclass="ORD"><mtext>&nbsp;</mtext><mi mathvariant="normal">m</mi></mrow><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">s</mi></mrow><mo>;</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">h</mi></mrow><mo>=</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>34</mn></mrow></msup><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">Js</mi></mrow></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">m</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">e</mi></mrow></mrow></msub><mo>=</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>31</mn></mrow></msup><mrow data-mjx-texclass="ORD"><mtext>&nbsp;</mtext><mi data-mjx-auto-op="false">kg</mi></mrow></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi>λ</mi><mo>=</mo><mfrac><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">h</mi></mrow><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">mv</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>34</mn></mrow></msup></mrow><mrow><mn>9</mn><mo>×</mo><msup><mn>10</mn><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>31</mn></mrow></msup><mo>×</mo><mn>1000</mn></mrow></mfrac><mo>=</mo><mn>666.67</mn><mo>×</mo><msup><mn>10</mn><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>9</mn></mrow></msup><mrow data-mjx-texclass="ORD"><mtext>&nbsp;</mtext><mi mathvariant="normal">m</mi></mrow></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mn>666.67</mn><mrow data-mjx-texclass="ORD"><mtext>&nbsp;</mtext><mi data-mjx-auto-op="false">nm</mi></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container>,

(B) is False : The characteristic of electrons emitted is independent of the material of the electrodes of the cathode ray tube.

(C) is True: In a cathode ray tube, electrons are emitted from the cathode and move towards the anode under the influence of an electric field.

(D) is False: The nature of the emitted electrons does not depend on the nature of the gas present in the cathode ray tube, but rather on the material of the electrodes and the potential difference applied across them.

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