Video Transcript
Which of the following is a correct equivalent formula for 𝛽 sub 𝑒 in terms of 𝛼
sub 𝑒? (A) 𝛼 sub 𝑒 divided by one minus 𝛼 sub 𝑒. (B) One divided by 𝛼 sub 𝑒. (C) 𝛼 sub 𝑒 divided by one plus 𝛼 sub 𝑒. (D) One minus one over 𝛼 sub 𝑒.
In this question, we are being asked about the quantities 𝛼 sub 𝑒 and 𝛽 sub 𝑒,
which are terms describing current gain in a transistor. Let’s recall that a transistor is an electric component made of a collector, a base,
and an emitter. Current through these parts is labeled 𝐼 sub 𝐶, 𝐼 sub 𝐵, and 𝐼 sub 𝐸,
respectively. These currents relate to one another via the equations 𝐼 sub 𝐸 equals 𝐼 sub 𝐶
plus 𝐼 sub 𝐵 and 𝐼 sub 𝐶 equals 𝐼 sub 𝐸 times 𝛼 sub 𝑒. The first equation relates the magnitudes of the currents. The second equation relates the collector and emitter currents to the constant of
proportionality 𝛼 sub 𝑒.
Now, we are being asked here to relate 𝛼 sub 𝑒 to 𝛽 sub 𝑒, so let’s recall how 𝛽
sub 𝑒 is defined. 𝛽 sub 𝑒 is the ratio of the collector current to the base current, 𝐼 sub 𝐶 over
𝐼 sub 𝐵. Our approach to solving this question will be to rearrange and combine these two
equations to get expressions for 𝐼 sub 𝐶 and 𝐼 sub 𝐵 involving the quantity 𝛼
sub 𝑒 that we can then substitute into this equation. We’ll begin by taking this first equation and subtracting 𝐼 sub 𝐶 from both sides,
which gives us an expression for 𝐼 sub 𝐵 in terms of 𝐼 sub 𝐸 and 𝐼 sub 𝐶.
We can now use this second equation to substitute 𝐼 sub 𝐸 times 𝛼 sub 𝑒 in place
of 𝐼 sub 𝐶 in this equation. We then have that 𝐼 sub 𝐵 is equal to 𝐼 sub 𝐸 minus 𝐼 sub 𝐸 times 𝛼 sub
𝑒. We can now factor out the 𝐼 sub 𝐸 that appears in both terms on the right-hand
side, giving us the equation 𝐼 sub 𝐵 is equal to 𝐼 sub 𝐸 multiplied by one minus
𝛼 sub 𝑒. We now have this equation which relates 𝐼 sub 𝐵 to 𝛼 sub 𝑒 and 𝐼 sub 𝐸, in
addition to this equation relating 𝐼 sub 𝐶 to the same two quantities 𝛼 sub 𝑒
and 𝐼 sub 𝐸.
We can now use these two equations in order to substitute for 𝐼 sub 𝐶 and 𝐼 sub 𝐵
in this equation for 𝛽 sub 𝑒. Let’s clear some space on screen to do this. Subbing in our expressions for 𝐼 sub 𝐶 and 𝐼 sub 𝐵 into our equation for 𝛽 sub
𝑒, we find that 𝛽 sub 𝑒 is equal to 𝐼 sub 𝐸 times 𝛼 sub 𝑒 divided by 𝐼 sub
𝐸 times one minus 𝛼 sub 𝑒. We can notice that there’s an 𝐼 sub 𝐸 term in the numerator and denominator of this
expression, which will cancel each other out. This leaves us with an expression for 𝛽 sub 𝑒 in terms of 𝛼 sub 𝑒, which is what
we were asked to find.
We have that 𝛽 sub 𝑒 is equal to 𝛼 sub 𝑒 divided by one minus 𝛼 sub 𝑒. This matches the expression given in answer choice (A). And so we can identify option (A) as being the correct answer. 𝛽 sub 𝑒 is equal to 𝛼 sub 𝑒 divided by one minus 𝛼 sub 𝑒.
