Question

A copper rod of length 0.77 m is lying on a frictionless table (see the drawing). Each end of the rod is attached to a fixed wire by an unstretched spring that has a spring constant of k = 76 N/m. A magnetic field with a strength of 0.14 T is oriented perpendicular to the surface of the table.

(a) What must be the direction of the current in the copper rod that causes the springs to stretch?
The current flows (left-to-right/right-to-left) in the copper rod.

(b) If the current is 14 A, by how much does each spring stretch?
________ m

Answer 1

The direction of the current in the copper rod that causes the springs to stretch is left-to-right. Each spring will stretch by approximately -0.384 m when a current of 14 A flows through the copper rod.

Step-by-step explanation:

To stretch the springs, the current in the copper rod must be directed from left-to-right. This is because when a current-carrying wire is placed in a magnetic field, a force is exerted on the wire due to the interaction between the magnetic field and the current. According to Fleming's left-hand rule, when the thumb points in the direction of the current, the fingers curl in the direction of the magnetic field.

To find the amount by which each spring stretches, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement of the spring from its equilibrium position.

The formula for Hooke's Law is: F = -kx

where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring.

Since each end of the copper rod is attached to a spring, the displacement of each spring is the same. Therefore, we can write:

F = -kx1

F = -kx2

where x1 and x2 represent the displacement of each spring.

Since the magnitude of the force exerted by the spring is equal to the force exerted by the magnetic field on the current-carrying wire, we can equate these two forces:

F = BIl

where B is the magnetic field strength, I is the current, and l is the length of the wire.

Substituting the expressions for the force exerted by the spring and the force exerted by the magnetic field, we get:

-kx1 = BIl

-kx2 = BIl

Solving these equations for x1 and x2, we find:

x1 = -(BIl)/k

x2 = -(BIl)/k

Since the displacement of each spring is the same, we can calculate the total displacement by summing x1 and x2:

x_total = x1 + x2 = 2x1 = 2x2 = -(2BIl)/k

Substituting the given values, we have:

x_total = -(2*0.14*14*0.77)/76 = -0.384 m

Answer 2

a) The direction of the current in the copper rod that causes the springs to stretch is left-to-right. This is because the magnetic field is oriented perpendicular to the surface of the table and the force exerted on the rod by the magnetic field is given by F = qVB, where F is the force, q is the charge of the particle, V is the velocity of the particle, and B is the strength of the magnetic field. Since the force is perpendicular to the velocity of the particle, the direction of the current that produces this force must be parallel to the magnetic field. In this case, the magnetic field is oriented perpendicular to the surface of the table, so the current must flow in the direction left-to-right to produce a force that stretches the springs.

b) To find by how much each spring stretches, you can use the formula F = kx, where F is the force exerted by the spring, k is the spring constant, and x is the distance the spring stretches. In this case, the force exerted by the spring is given by F = qVB, and the spring constant is k = 76 N/m. The charge of the particle is q = Ie, where I is the current and e is the charge of an electron. The velocity of the particle is V = L / t, where L is the length of the rod and t is the time it takes the particle to travel the length of the rod. Since the current is 14 A and the length of the rod is 0.77 m, the velocity of the particle is V = 0.77 m / (1 / 14 A) = 10.78 m/s. The strength of the magnetic field is 0.14 T, so the force exerted by the spring is F = (14 A)(1.60 x 10^-19 C)(10.78 m/s)(0.14 T) = 2.40 x 10^-17 N. Therefore, the distance each spring stretches is x = F / k = 2.40 x 10^-17 N / 76 N/m = 3.16 x 10^-19 meters.