A soap bubble of initial radius R is to be bl | vedclass

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(D) soap bubble has two surfaces (inner and outer), so the surface area is $2 	imes 4 \pi r^2 = 8 \pi r^2$.Initial radius $r_i = R$.Initial surface e

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$24$

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(D) soap bubble has two surfaces (inner and outer), so the surface area is $2 	imes 4 \pi r^2 = 8 \pi r^2$.Initial radius $r_i = R$.Initial surface energy $U_i = T 	imes (8 \pi R^2) = 8 \pi R^2 T$.Final diameter is doubled, so final radius $r_f = 2R$.Final surface energy $U_f = T 	imes (8 \pi (2R)^2) = T 	imes (8 \pi 	imes 4R^2) = 32 \pi R^2 T$.The surface energy needed is $\Delta U = U_f - U_i$.$\Delta U = 32 \pi R^2 T - 8 \pi R^2 T = 24 \pi R^2 T$.

$A$ soap bubble of initial radius $R$ is to be blown up. The surface tension of the soap film is $T$. The surface energy needed to double the diameter of the bubble is (in $\pi R^2 T$)

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