MHT CET · Physics · Magnetic Effects of Current

An alternating electric field of frequency $f$ is applied across the radius $R$ of a cyclotron to accelerate protons (mass $m$ ). The operating magnetic field $B$ used and K.E. of the proton beam produced by it are respectively ( $e =$ charge on proton)

A $\frac{2 π m f}{e}, 2 π^{2} m f^{2} R^{2}$
B $\frac{2 π^{2} m f}{e^{2}}, 4 π^{2} m f^{2} R^{2}$
C $\frac{π^{} m f}{e^{}}, π^{2} m f^{2} R^{2}$
D $\frac{2 π^{2} m^{2} f^{2}}{e^{}}, 2 π^{2} m^{2} f^{2} R^{2}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{2 π m f}{e}, 2 π^{2} m f^{2} R^{2}$

Step-by-step Solution

Detailed explanation

T = \frac{2 π m}{e B}

R = \frac{m f}{e B} = \frac{P}{e B}

B = \frac{2 π m}{e} f

P = e B R = e \times \frac{2 π m υ}{e} R

K . E = \frac{P^{2}}{2 m} = \frac{{(2 π m f R)}^{2}}{2 m}

= 2 π^{2} m f^{2} R^{2}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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An alternating electric field of frequency f is applied across the radius R of a cyclotron to accelerate protons (mass m ). The operating magnetic field B used and K.E. of the proton beam produced by it are respectively ( e= charge on proton)

Step-by-step Solution

An alternating electric field of frequency $f$ is applied across the radius $R$ of a cyclotron to accelerate protons (mass $m$ ). The operating magnetic field $B$ used and K.E. of the proton beam produced by it are respectively ( $e =$ charge on proton)