Two charges of \(-4 \mu \mathrm{C}\) and \(+4 \mu \mathrm{C}\) are placed at the points \(\mathrm{A}(1,0,4) \mathrm{m}\) and \(\mathrm{B}(2,-1,5) \mathrm{m}\) located in an electric field \(\overrightarrow{\mathrm{E}}=0.20 \hat{\mathrm{i}} \mathrm{V} / \mathrm{cm}\). The magnitude of the torque acting on the dipole is \(8 \sqrt{\alpha} \times 10^{-5} \mathrm{Nm}\), Where \(\alpha=\) \_\_\_\_.

JEE Main · 2024 · 27 Jan 2024 · Shift 2 · Electrostatics

Two charges of

- 4 μ C

and

+ 4 μ C

are placed at the points

A (1, 0, 4) m

and

B (2, - 1, 5) m

located in an electric field

E = 0.20 \hat{i} V / cm

. The magnitude of the torque acting on the dipole is

8 α \times 1 0^{- 5} Nm

, Where

α =

____.

Answer: 2

The dipole moment points from the negative to the positive charge:

ℓ = (2 - 1) \hat{i} + (- 1 - 0) \hat{j} + (5 - 4) \hat{k} = \hat{i} - \hat{j} + \hat{k}

m, so

p = q ℓ = 4 \times 1 0^{- 6} (\hat{i} - \hat{j} + \hat{k}) C\cdotm .

The field is

E = 0.20 V / c m = 20 \hat{i} V / m

. The torque is

τ = p \times E = 4 \times 1 0^{- 6} (\hat{i} - \hat{j} + \hat{k}) \times 20 \hat{i} = 80 \times 1 0^{- 6} (\hat{j} + \hat{k}) N\cdotm,

using

\hat{j} \times \hat{i} = - \hat{k}

and

\hat{k} \times \hat{i} = \hat{j}

. Its magnitude is

802 \times 1 0^{- 6} = 82 \times 1 0^{- 5} N\cdotm

. Matching this to

8 α \times 1 0^{- 5}

gives

α = 2

JEE Main · 2024 · 27 Jan 2024 · Shift 2 · Integer answer