Why Is Cosine Used to Calculate Induced Voltage in a Conductor?

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http://www.berkeleyscience.com/maxwells.htm

"""""When a conductor (a wire) is placed in an electric field, the field may induce a voltage in the conductor. The diagram shows an electric field in the x direction with magnitude E. If the conductor is perpendicular to the E field, there is no induced voltage. If the conductor is aligned with the E field, the induced voltage is E·l where l is the length of the wire. If the angle between the E field and the wire is θ then the induced voltage is E·cos(θ)·l."""

^^ this is quoted from the link i just gave ^^
It is with reference to this image here : http://www.berkeleyscience.com/images/efield.jpg

Now I am curious why Cos is used when determining the induced voltage, that's all i am curious about. Why not Sin? How would you determine to use Cos? Please can you help me in laymans terms because I am only beginning Trig properly

 

[Astronuc]

The maximum voltage is induced when the E-field vector points along the conductor. When the E-field is at some angle, then it is the 'projection' or component of the E-field vector parallel to the conductor that causes the voltage.

Think about - what is the significance of the cos and the angle between the hypotenuse and one leg.
http://hyperphysics.phy-astr.gsu.edu/hbase/ttrig.html

If one was interested in the projection or component of the E-field in the transverse direction to the conductor, then one would used the sin. But the transverse E-field would not be so significant here, although it would produce a tranverse voltage across the conductor, as a opposed to along the conductor.

 

[nlightner]

and I am finding it hard to understand.

I can explain why Cos is used in this situation.

Firstly, it is important to understand that the induced voltage in a conductor is dependent on the angle between the electric field and the conductor. This angle is represented by θ in the equation.

Now, let's consider the definition of Cosine (Cos) in trigonometry. Cosine is a ratio of the adjacent side to the hypotenuse in a right triangle. In this scenario, the adjacent side would be the length of the wire (l) and the hypotenuse would be the magnitude of the electric field (E).

When we multiply these two values with the angle (θ) between them, we get the induced voltage (E·cos(θ)·l). This is because the induced voltage is directly proportional to the length of the wire (l) and the magnitude of the electric field (E).

If we were to use Sin instead of Cos, we would get a different value for the induced voltage because Sin is a ratio of the opposite side to the hypotenuse. This would not accurately represent the relationship between the induced voltage and the angle between the electric field and the conductor.

Therefore, in this scenario, Cos is used because it accurately represents the relationship between the induced voltage, the length of the wire, and the magnitude of the electric field. I hope this explanation helps you understand why Cos is used in this situation.