The electric potential in a certain region is
V = a x2 + b x + c,
where a = 14 V/m2, b = −19 V/m, and c = 61 V.
Determine the position where the electric field is zero.
Answer in units of m
Please help!
ax²+bx+c=0,
14x²-19 x +61 =0,
x= [19±sqrt{19²+4•14•61}]/2•14 =
=(19±61.5)/28.
x₁=2.88 m,
x₂=-1.52 m
Oh, don't worry, I'm here to help! Let's find out where the electric field is zero.
To find the position where the electric field is zero, we need to determine the value of x in the equation V = a x^2 + b x + c when the electric potential V is zero.
So, let's set V equal to zero:
0 = a x^2 + b x + c
Now let's substitute the given values for a, b, and c:
0 = 14 x^2 - 19 x + 61
Hmm, this equation doesn't look too friendly for solving directly. So, let's summon the help of a quadratic formula! The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 14, b = -19, and c = 61.
Plugging in these values, we get:
x = (19 ± √((-19)^2 - 4 * 14 * 61)) / (2 * 14)
Calculating further, we get:
x = (19 ± √(361 - 3448)) / 28
x = (19 ± √(-3087)) / 28
Uh oh, it seems that we have a negative number under the square root, which means that the electric field is imaginary at this position.
So, in this specific region, there is no real position where the electric field is zero. It's a "no field zone" I suppose. Keep searching, and you'll find another region with a zero electric field.
I hope that helps, and remember, sometimes a little humor can lighten the electric potential of a situation!
To determine the position where the electric field is zero, we need to find the value of x when the electric potential V is zero.
Given:
V = a x^2 + b x + c
a = 14 V/m^2
b = -19 V/m
c = 61 V
When the electric field is zero, the potential V will also be zero. So, we can set V = 0:
0 = 14 x^2 - 19 x + 61
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the given values, we have:
x = (-(-19) ± √((-19)^2 - 4 * 14 * 61)) / (2 * 14)
x = (19 ± √(361 - 3508)) / 28
x = (19 ± √(-3147)) / 28
Since we obtained a negative value inside the square root, this means that there are no real solutions to the equation. Therefore, there is no position where the electric field is zero in this particular situation.
To determine the position where the electric field is zero, we first need to find the electric field expression. The electric field (E) is related to the electric potential (V) by the equation:
E = -dV/dx
Where dV/dx represents the derivative of V with respect to x.
In this case, the electric potential function is given by V = a*x^2 + b*x + c, where a = 14 V/m^2, b = -19 V/m, and c = 61 V.
To find the electric field expression, we need to take the derivative of V with respect to x:
dV/dx = 2*a*x + b
Now that we have the electric field expression, we can find the position where the electric field is zero by setting E = 0 and solving for x:
0 = 2*a*x + b
Substituting the values of a and b we have:
0 = 2*(14 V/m^2)*x - 19 V/m
Solving for x, we get:
2*(14 V/m^2)*x = 19 V/m
x = 19 V/m / (2*(14 V/m^2))
x = 19 V/m / 28 V/m^2
x = 19/28 m
Therefore, the position where the electric field is zero is x = 19/28 m.