Identify the incorrect relationship

Options

A. Number of waves in an orbit, `n=(2pir)/lambda`
B. Number of revolutions of an electron per second in `n^"th"` orbit =`V_n/(2pir_n)`
C. Wavelength of an electron =`h/p`
D. Speed of a (de-Broglie wavelength ) particle accelerated by potential difference V is `v="2eV"/m`

Answer

To solve the question of identifying the incorrect relationship from the provided options, we will analyze each option step by step.

### Step 1: Analyze Option A
**Statement:** The number of waves in an orbit is given by \( n = \frac{2\pi r}{\lambda} \).

**Explanation:** According to Bohr's postulate, the circumference of an electron's orbit must be an integral multiple of the wavelength (\( \lambda \)). Thus, the number of wavelengths in the orbit is indeed given by the formula \( n = \frac{2\pi r}{\lambda} \), where \( r \) is the radius of the orbit.

**Conclusion:** This option is correct.

### Step 2: Analyze Option B
**Statement:** The number of revolutions of an electron per second in the nth orbit is given by \( \frac{v}{2\pi r} \).

**Explanation:** The number of revolutions per second (frequency) can be calculated by dividing the speed \( v \) by the circumference of the orbit \( 2\pi r \). Therefore, the formula for the number of revolutions per second is indeed \( \frac{v}{2\pi r} \).

**Conclusion:** This option is also correct.

### Step 3: Analyze Option C
**Statement:** The wavelength of an electron is given by \( \lambda = \frac{h}{p} \).

**Explanation:** According to de Broglie's hypothesis, the wavelength \( \lambda \) of a particle is inversely proportional to its momentum \( p \). The relationship is given by \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant.

**Conclusion:** This option is correct as well.

### Step 4: Analyze Option D
**Statement:** The speed of a particle accelerated by a potential difference \( V \) is given by \( v = \sqrt{\frac{2EV}{m}} \).

**Explanation:** The kinetic energy \( KE \) gained by a particle when accelerated through a potential difference \( V \) is given by \( KE = qV \). For an electron, this can be expressed as \( KE = \frac{1}{2} mv^2 \). Equating these gives us:

\[ \frac{1}{2} mv^2 = qV \implies v^2 = \frac{2qV}{m} \implies v = \sqrt{\frac{2qV}{m}}. \]

However, the option states \( v = \sqrt{\frac{2EV}{m}} \), which is incorrect because \( E \) should be replaced with \( q \) (the charge of the electron).

**Conclusion:** This option is incorrect.

### Final Answer
The incorrect relationship is **Option D**.

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